The Erwin Schrodinger International Boltzmanngasse

ESI Institute for A Wien Austria

Remarks on the Life and Research of

Roland L Dobrushin

RA Minlos

EA Pechersky

YuM Suhov

Vienna Preprint ESI Septemb er

Supp orted by Federal Ministry of Science and Research Austria

Available via httpwwwesiacat

Submitted to Dobrushins memorial volume of

Journal of Applied

and Stochastic Analysis

REMARKS ON THE LIFE AND RESEARCH OF

ROLAND L DOBRUSHIN

RAMinlos EAPechersky and YuMSuhov

Abstract The life and research work of Professor RL Dobrushin had

a profound inuence on several areas of Probability Theory and

Mathematical Physics The pap er contains a biographical note a review of Dobrushins

results and the list of his publications

A FOREWORD

In the autumn of the mathematics community was shaken by the news from

that Roland Dobrushin had died of cancer on th Novemb er aged He was at the

p eak of his creative p owers a great numb er of his pap ers were in print or preparation

It is imp ossible to know how many other works were in conception we hop e that his

colleagues and pupils will b e able to reconstruct at least some of his ideas Dobrushins

energy during the last p erio d of his life is also demonstrated by the extensive travel

he underto ok in when he was already seriously ill among other meetings he

attended the Conference in Mathematical Physics Aragats Armenia May the

rd Bernoulli Conference on Sto chastic Pro cesses and Their Applications Singap ore

June and the Probability and Physics Conference Renkum The Netherlands

August he was planning to sp end the autumn of in the Schrodinger Institute

in Vienna collab orating with a numb er of coworkers

It is dicult to assess the scale of the loss to mathematics in general and Russian

mathematics in particular o ccasioned by his death Dobrushins enormous contribution

to mo dern mathematics is not conned to his publications he was a man who generated

a sp ecial mathematical aura Everyb o dy within his orbit who had the slightest talent

The Dobrushin Mathematical Lab oratory Institute for Problems of Information

Transmission The Russian Academy of Sciences Bolshoi Karetnyi Per GSP

Moscow Russia

Statistical Lab oratory DPMMS University of Cambridge Mill Lane Cambridge

CB SB England UK

This work has b een supp orted in part by the American Mathematical So ciety the

International Science Foundation Grant NME the EC Grant Training Mobility

and Research under Pro ject No Contracts CHRXCT and ERBFMRX

CT and the INTAS Grant under Pro ject Mathematical Metho ds for Sto chastic

Discrete Event Systems INTAS

for creating new mathematical results was quickly included in active and absorbing

research Such research was always deeply motivated imp ortant for newcomers and

conducted to the highest standards For many mathematicians the sub ject they b egan

to work on with Dobrushin b ecame their main topic of fruitful research for years if not

decades to come His ideas and views like waves in water p ercolated and continue

p ercolating throughout the mathematical community not always recognised as initiated

by Dobrushin Alas the source of the waves is no longer with us

A numb er of events dedicated to Dobrushins memory have taken place or are

planned eg a session of The Mathematical So ciety April the con

ferences at The Schrodinger Institute Vienna Septemb er and INRIA

VersailleRo cquencourt Octob er Obituaries and biographical articles

have b een published a numb er of journals are to have sp ecial

issues in his memory The present pap er is an attempt to describ e some of his research

contributions we have tried to make the material accessible to a large probabilistic au

dience maintaining at the same time the necessary level of mathematical rigour We

pay sp ecial attention to the origins of his main ideas and to a retrosp ective analysis of

his metho ds we b elieve that these are imp ortant issues that have p erhaps not b een

discussed in detail in the literature so far A brief biography is provided where we fo cus

on several asp ects of his life Dobrushins p ersonality had a huge impact on entire elds

of research in Russia and abroad We understand that our comments are inevitably

onesided and selective it is imp ossible within the limits of a single article to analyse

in depth his inuence up on the mo dern state of research

We also give a complete list of Dobrushins published works In the case of Do

brushins pap ers originally published in Russian and ocially translated to English we

refer to year of the Russian publication

The references to the translated Russian pap ers by other authors are to their English

translation In general while referring to the Russian pap ers volumes the names of

the authors and the titles of the journals p erio dicals and volumes are repro duced in

the Russian transliteration whereas the titles of the pap ers are given in the English

translation We ap ologise to the reader for p ossible divergency with other translated

versions of of the same Russian titles which may exist in the literature

Commenting on the pap ers in which Dobrushin was a coauthor we give his name

only for which we ap ologise to his numerous coworkers this is merely for the sake

of unity of style However it should b e noted that at least in our exp erience he was

always the natural leader of a team without b eing patronising His ideas almost always

worked well and his picture of the nal result was astonishingly correct

A BIOGRAPHICAL NOTE

Dobrushin who was of German Jewish and Russian origin was b orn on July in

Leningrad now St Petersburg His parents died when he was a child and he was brought

up by relatives in Moscow His mathematical abilities were noted at scho ol but it is not

known whether his scho ol interests were conned to mathematics However it is a fact

that he successfully to ok part in mathematical olympiads a p opular comp etition op en

to talented scho olchildren in which they had to solve sp ecially selected and prepared

questions the term olympiad problem in Russian mathematical jargon describ es a

particular style of question at these comp etitions A rememb ered episo de o ccurred

when in the course of solving an olympiad problem Dobrushin had to use an axiom

ab out a partition of a plane by a single line which he was not aware of at the time As

a result he wrote in his solution that much to his embarrassment he did not know what

a straight line was a remark noted by the examiners

After nishing secondary scho ol in Dobrushin applied for admission to the

Department of Physics FizFak of MSU However he failed

to pass the entrance examination although apparently not on the basis of his abilities

or knowledge of the sub jects This was during a p erio d of rising antisemitism in ocial

propaganda the Soviet authorities were particularly sensitive ab out admitting Jews to

this department where a large numb er of future nuclear scientists were trained

Nevertheless he was able to gain admission to the Department of Mechanics and

Mathematics MekhMat of MSU From the b eginning he to ok an active part in a

student seminar series run by Dynkin Here he acquired a deep interest in probability

theory and a particular probabilistic style of thinking which often distinguishes great

scientists in the eld After graduating in Dobrushin was admitted as a research

student with Kolmogorov as sup ervisor Once again he had great diculty in obtaining

this studentship for reasons unrelated to his research It is well known that Kolmogorov

had to use all his inuence to have him admitted Many excellent mathematicians who

graduated at MSU at around the same time did not get research studentships

In Dobrushin completed and defended his PhD thesis A Lo cal Limit Theorem

for Markov Chains he was then given a p osition at the Probability Section of Mekh

Mat In he was awarded the prize of the Moscow Mathematical So ciety for young

mathematicians a prestigious award though mo dest in material terms that marked

many future celebrities of Soviet mathematics Dobrushins thesis improved a series

of theorems of his predecessors among whom one can mention Markov Bernstein and

Linnik

The s were a time when Information Theory emerged and quickly progressed

following the works of Shannon Dobrushin also b ecame interested in this area We

can only guess what moved him in this direction such a decision might have b een in

uenced by Kolmogorov who advised young mathematicians to work in the new elds

of Probability Theory However one could supp ose that Dobrushin was attracted by a

striking critical p oint phenomenon discovered by Shannon ab out the error probability

in deco ding a long message Dobrushin studied general conditions under which such a

phenomenon holds as b efore he found a concept that is essential for the validity of

Shannons theorems the socalled information stability Working on these problems

he sp ent a large part of his time on propagating the ideas and metho ds of Information

Theory he always to ok seriously the task of p opularising new ideas and was indefatiga

ble in this capacity He edited the Information Theory section of Soviet Mathematical

Reviews and b egan running a seminar series in the recently created Institute for Prob

lems of Information Transmission IPIT of the then Soviet now Russian Academy of

Sciences He ran this seminar series until his nal days and to ok his duties extremely

seriously In he prepared and defended his do ctorate on his results from Shannons

theory His do ctorate was awarded at the Moscow Institute of Applied Mathematics of

the AS where the mathematical part of the Soviet space programme was develop ed at

that time

In the early s Dobrushin felt that the sub ject of Information Theory was b egin

ning to b e exhausted although he continued with some interruptions publishing pap ers

in this eld until the late s According to his colleagues and friends he had similar

feelings ab out many areas of classical Probability Theory Conceding that the whole

stream of works in classical directions rich in results and traditions served an imp ortant

purp ose in constructing a unied theory he came to the conclusion that fo cusing on tra

ditional approaches somehow slowed down the development of completely new elds He

gave much thought to this problem and voiced his dissatisfaction with the situation to

his colleagues His frequent conversations with one of the authors of this pap er RAM

were directed towards fundamental questions of in particular the

problem of phase transitions In general his intention was to nd common ground b e

tween Physics and Probability Theory recall his attempt to b e admitted to FizFak of

MSU

The second half of the s and the b eginning of the s saw the start of a p o

litical thaw which however incomplete and contradictory irreversibly changed p eoples

outlo ok and created a spirit of indep endence and in many cases deance of ocial do c

trines The future dissident movement was founded in this spirit of deance as well as

a general nonconformist attitude widely p opular among scientists writers painters and

other memb ers of the intelligentsia esp ecially in Moscow and Leningrad However the

regime was still a p owerful structure and it had many supp orters who for one reason

or another were prepared to opp ose changes and close their eyes to repressive measures

against those critics of the system who dared to go to o far Dobrushin had a so cially

active mind and a very strong and indep endent p ersonality Together with his deep

conviction that demo cratic principles should b e intro duced into Russian so ciety this

inevitably put him on a collision course with ocialdom and its supp orters The story

of his confrontation with the huge repressive machine is worth a separate article here

we mention a few facts only

In autumn a group of MekhMat students made public a few copies of a

typ ewritten literary bulletin An early example of samizdat there were among its au

thors and distributors names that left their mark on the future development of Russian

mathematics From a contemp orary viewp oint the bulletins contents were inno cuous

They included a sp eech by a p opular soviet writer in which he criticised several of his

colleagues hiding an obvious lack of talent b ehind the ortho doxy of so cialist realism

excerpts from John Reeds essay on Trotsky who till the Fall of was considered

a p olitical evil of the Soviet history and a numb er of verses by young p o ets denied

publication in the tightly controlled ocial magazines The authors of the bulletin were

p erhaps naive in thinking that the time of longawaited freedom had arrived

The reaction of the MekhMat authorities was nervous The Soviet Army had just

invaded Hungary to crush reform and there was a danger of confrontation in the Middle

East over Nassers nationalisation of the Suez canal In this situation it was decided

that the bulletin should b e treated as an inside enemys activity It should b e noted

that some prominent MekhMats memb ers were outraged by the bulletin primarily not

b ecause they disagreed with its contents or with the fact of its distribution but b ecause

they were afraid of ocial repressions against the department

The departmental authorities summoned a meeting of the sta and students with

the declared ob jective of condemning and punishing the moral mutineersbut in

reality to demonstrate unanimous approval of the ocial line at this complicated p erio d

Such meetings were part of a long tradition in Soviet p olitical life and their scenario

was tested and improved countless times although in p ostStalin times enthusiasm for

condemnation was not as great as it had b een Sp eakers carefully selected by the

organisers in advance duly denounced with various degrees of histrionic severity the

heretics and some of the accused demonstrated various degrees of rep entance However

the planned course of the meeting was disrupted by Dobrushins sp eech in which he

declared that the bulletin was a manifestation of the eternal principles of freedom and

the authorities could only b enet if everyb o dy were free to sp eak their mind The

absurdity of the pro ceedings immediately b ecame clear to all present and the authorities

to their outrage lost control of the meeting

However Dobrushins words cost him and not only him dearly The lo cal Com

munist Party b osses insisted that he should b e red This was opp osed by Kolmogorov

but even his inuence had limits There was no way for Dobrushin to b e promoted and

he was not p ermitted to travel abroad except to some countries under Soviet domina

tion The system invented a sp ecial term for such a category of p eople very lo osely

it can b e translated as nonvoyager or more precisely voyager to so cialist countries

exclusively All this however could not prevent him continuing to defy the authorities

he was a cosignatory of almost all the protest letters that circulated in Moscow in the

s

With time the repressive side of the regime prevailed and Dobrushins p osition

at MSU b ecame precarious At the b eginning of he left MekhMat and accepted

an oer from the Institute for Problems of Information Transmission He organised a

lab oratory at IPIT and worked as its head until his death The main direction of research

in his lab oratory was initially Information and Co ding Theory he later added the Theory

of Complex Sto chastic Systems which embraced his emerging interests in Statistical

Mechanics and Queueing Network Theory see b elow He also taught parttime at the

Moscow Physical and Technical Institute FizTekh where he held a professorship from

to He to ok an active part in editing Problems of Information Transmission

which under him b ecame a wellknown and resp ected journal It must b e said that the

leadership of IPIT showed great courage in giving him such a prominent p osition partly

explained by the dierent atmosphere prevailing in many institutes of the Academy of

Sciences There was traditionally a strong lib eral spirit in these places and numerous

dissidents and refusniks enjoyed the loyalty of colleagues and the administration There

was also a general inertia of the machinery of repression which made for greater freedom

As the head of the lab oratory Dobrushin showed an extraordinary ability to recruit

talented young mathematicians and direct their work in a wide variety of problems

The climate he created was extremely favourable for genuine intensive research and

encouraged mutual sympathy and friendship b etween the sta Despite its relatively

small size ab out ten p eople the lab oratory achieved prominence in several elds of

mathematics One memb er was awarded a Fields Medal another received the prize of

the Europ ean Union of Mathematicians a third a distinguished prize of the IEEE In

general Dobrushins presence always created an atmosphere of go o d spirits a desire to

learn and pro duce new work and a readiness to help and share with others

The p erio d from the mids was the golden age of Dobrushins research career

Without interrupting his work in Information Theory in he together with Min

los intro duced a seminar series at MSU with the general aim of bringing Statistical

Mechanics into the context of probability theory The following year they are joined by

Sinai and for a short p erio d by Berezin and Schwartz and later still Malyshev The

seminar series on Statistical Physics b ecame a forum for intensive discussion of various

In this context the terms Statistical Physics and Statistical Mechanics are inter

problems in the new eld and quickly gained an international reputation A numb er of

essential probabilistic concepts and constructions were created here which describ e the

phenomena of Statistical Mechanics Dobrushins main achievements in were

the concepts of a sp ecication and of a Gibbs random eld He understo o d that one

of the most imp ortant phenomena of interest in Statistical Physics phase transition is

describ ed as a nonuniqueness of a Gibbs eld with a given sp ecication He then gave

a short and b eautiful pro of of the existence of phase transitions in the Ising mo del and

its mo dications in dimensions two and higher and went further by investigating the

structure of the set of pure phases in these mo dels His main results in this direction are

published in b a ac a ab ab ab c

In our view these pap ers are imp ortant not only b ecause they laid the foundation of the

mo dern equilibrium Statistical Mechanics and solved a numb er of dicult problems but

also and p erhaps mainly b ecause they contain or lead to many op en questions which

we are sure will inspire future waves of research It can b e observed that many of his

later works inevitably b ecame technically much more involved and less accessible for a

wide audience

Dobrushins results of b ecame instantly famous and attracted crowds of

new researchers from across the world There were countless conferences and symp osia

and mutual visits where his theory was discussed at length and in detail However the

author himself was not able to put his fo ot b eyond the Iron Curtain although he was

inundated with invitations Instead the stream of scientists from the Benelux countries

France Germany Italy Japan Scandinavian countries Switzerland UK USA came to

see Dobrushin in Russia or Soviet blo ck states Dobrushins case b ecame a headache for

Soviet ocials but the system remained adamant A curious episo de o ccurred when the

pap ers ab out a planned visit of an American colleague to IPIT arrived in the institute

during a very busy time The visitors name was missp elled in the course of translation

to Russian and Dobrushin in hurry did not recognise him Consequently the institute

did not give its approval and the visit was cancelled In traditions of Cold War the

real cause of cancelling the visit was not made known to the American colleague who

susp ected that the Soviet repressive institutions prevented Dobrushin not only from

going abroad but also from meeting foreigners in his own country such measures were

sometimes applied to p eople who fell in disgrace under the Soviet regime The colleague

gave an interview to Voice of America which was subsequently broadcasted to the Soviet

Union In these days the broadcasts from the West were regularly listened to by many

in USSR it was an alternative source of news to the ocial Soviet media Learning

ab out the broadcast Dobrushin rememb ered the case It to ok another year to make the

visit p ossible but at the end everyb o dy was happy

Working in an institute which sp ecialised in the study of various asp ects of infor

mation transmission Dobrushin naturally continued his own interest in these areas By

the mids his attention was mainly fo cused on problems in Queueing Network Theory

Here the ob ject of study is a collection of servers that pro cess a ow of tasks which

dep ending on the context may b e messages calls programs etc according to certain

rules the problems lie in assessing delays in pro cessing the tasks loss probabilities non

overload conditions etc Dobrushin approached these problems by using analogies with

ob jects from Statistical Physics His inuence in this eld went far b eyond his published

works and may b e traced in numerous pap ers by his followers

changeable

From when the Soviet Union entered the nal phase of p erestroika Dobrushin

was allowed to travel without restrictions With the change of p olitical regime he was also

accepted back at Moscow State University from to his death he held a parttime

professorial p osition at the Probability Section of MekhMat In general the character

of research in the USSR in many elds of mathematics and theoretical physics changed

dramatically at that time The numb er of visiting scientists from the West went down

whereas the opp osite stream of visitors from the Soviet Union b ecame much more intense

The numb er of trips abroad and their duration were considered by many as a sign of

reputation and b ecame a matter of comp etition The worsening economic and so cial

situation forced the emigration temp orary or p ermanent of leading and prominent

sp ecialists in practically all elds Famous Moscow seminar series went through hard

times and many of them ceased altogether This was the case with the seminar series in

Statistical Physics it continued with interruptions until and was then terminated

In this situation Dobrushin was one of the few whose enthusiasm remained constant

he was a profound optimist by nature Despite numerous oers he never sought a

p ermanent p osition in the West although in he accepted an invitation to sp end up

to six months a year in the Schrodinger Institute in Vienna He travelled widely but

was always glad to return to Moscow He loved the city and the country whatever it was

called and whichever p olitical force was in p ower After a p erio d of protest in the s he

was not directly involved in any p olitical activity but continued to b e deeply interested

in p olitics b oth inside Russia and abroad He was a great reader of journals and general

and p olitical magazines eg during the Soviet era he regularly read Marxist magazines

printed by leftist parties and groups in the West which he at some risk managed to

obtain from foreign friends and colleagues and kept in his at He evidently had a

very go o d understanding of the disp osition of p olitical forces his predictions of p olitical

events were always amazingly accurate

Dobrushins academic career at home develop ed in line with his status as a non

voyager The Mathematics Section of the USSR and then Russian Academy of Sciences

did not elect him either a full memb er or a memb ercorrespondent his candidature was

not even seriously discussed during election campaigns Despite his fame and reputation

he was treated as an outsider by the Soviet mathematical ocialdom as were many

other outstanding mathematicians of his time Partly this was due to antisemitism

partly to the servility towards the Soviet system of some Academy memb ers and partly

to the internal rivalry b etween dierent groups of academicians Dobrushins own anti

establishment attitude did little to help him to b ecome more p opular with the Soviet

academic elite His staunch reformist convictions were once more demonstrated in his

address to the General Assembly of the USSR AS in March at the high time of

p olitical debates on the future of the Soviet system in general and the particular role of

the Academy This sp eech was enthusiastically greeted by the large part of the audience

which included young researchers but was met with scepticism by the conservative part

of academicians world

In Dobrushin was elected an Honorary Memb er of the American Academy

of Fine Arts and Sciences in Boston High ocials of the Soviet Academy of Sciences

urged him to decline the honour it was the p eak of the last p erio d of confrontation of

sup erp owers but Dobrushin refused to follow their advice In he was elected

Asso ciated Foreign Memb er of the USA National Academy and in Memb er of the

Europ ean Academy

Dobrushin severd as the memb er of editorial or advisory b oards of Communications

in Mathematical Physics Journal of Statistical Physics Theory of Probability and Its

Applications and Selecta Mathematica Sovietica He also edited a numb er of volumes of

research pap ers of Russian authors b oth in Russian and English

From Dobrushin increased the numb er of sta in his lab oratory at IPIT and

greatly extended its research It is now called the Dobrushin Mathematical Lab oratory

and carries out research in a numb er of directions into Information and Co ding Theory

Queueing Network Theory Mathematical Physics and Representation Theory

A SURVEY OF DOBRUSHINS RESEARCH HISTORY

Markov pro cesses In the rst series of his published pap ers Dobrushin stud

ied nonhomogeneous Markov chains The main problem he was interested in is the

central limit theorem CLT for this class of pro cesses As mentioned ab ove he in

vented a sp ecic parameter known as the ergo dicity co ecient which describ es a degree

of homogeneity of a general Markov chain Consider an array of random variables

X X X

X X X

n n n

X X X

n n n

Each row X X X forms a Markov chain with the transition probabilities

n n n

P a A P X AjX a The ergo dicity co ecient of the nth row is

n

t t

t

dened by

n n

b A a A P P sup min

n

t t

tn

abA

Dobrushin showed that

If the random variables in are uniformly b ounded and

n

n

then for array the CLT holds true

This result concluded a series of pap ers op ened by a pap er by Markov who

proved the CLT for twostate chains under the condition

n

The next step was made by Bernstein who subsequently in and

obtained the CLT still in the case of two states when

n a

n

n b

n

and

n

n

for any In Sap ogov obtained the result where the chain again has two

states and ob eys For a general statespace the CLT was proved under condition

a in and b in For an arbitrary nite statespace it was proved under

by Linnik in Dobrushin ac proved the CLT under condition for a

general b ounded statespace eg an interval of a real line

Dobrushins result may b e treated as nal or close to nal b ecause Bernstein

gave an example with n where the CLT do es not hold Dobrushin himself

n

constructed an example where the statespace of the Markov chain is the real line and

the limiting law is not Gaussian but stable It is worth noting that the previous

statements were less elegant since they did not use directly the ergo dicity co ecient As

may b e seen from the denition assesses how dierent the probabilities of transition

n

are from dierent states into the same set the CLT holds when such a dierence is not

to o large In the pro of of the CLT Dobrushin uses martingales at that time such an

approach was yet not widely p opular The idea used his pap ers may b e describ ed by the

word contractivity in one form or another it was successfully used many times in his

subsequent pap ers

Apart from the CLT Dobrushin studied other problems from the Markov Chains

Theory He obtained necessary and sucient conditions for the numb er of jumps of a

nonhomogeneous Markov pro cess to b e nite a He constructed an example of

a Markov pro cess with countably many states each of which is instantaneous d

In a short note b he proved asymptotical normality of the time that a symmetric

random walk sp ends in a subset of Z Pap er e was a harbinger of his future

d

interests Let b e a translationinvariant random p oint eld in R with a nite rate

Assume that during the unit time each particle p erforms a random jump and the

d

displacements of dierent particles are I ID with probability density px x R Then

the pro cess at time n converges weakly to a homogeneous Poisson p oint eld

n

with rate

At that time pap er e seemed rather atypical and did not give rise to systematic

work in this direction However the concept of an innite particle system that emerged

from this pap er had serious impact on his later research From the end of the s

onwards Dobrushin fo cused on problems of information theory

Information Theory One of the main problems of co ding theory is given a

noisy channel of information transmission enco de a text at the input p ort and deco de it

at the output p ort so that the error probability b ecomes negligible or at least minimal

More precisely supp ose that a total of M dierent messages is given which are to b e

transmitted through a channel To each message one assigns a distinct co deword that is

n

a sequence of n binary digits M The collection M M of the co dewords is

n

n n

called a co de of length n it is a subset of the set f g of p ossible binary nwords

In the course of transmission random errors o ccur to start with we assume that

the statistics of the channel is known ie we know for any n and any a M and

n n

b f g the probability P a j b P a j b to receive a word a given that a

co deword b has b een sent A p opular example is a memoryless channel where P a j

Q

k

b w a j b a a a b b b a b or here w a j b is

i i n n i i i i

i

the probability of receiving binary symb ol a when b has b een sent

i i

Supp ose we use the maximum likeliho o d principle for deco ding according to which

the received word a is deco ded by the co deword b that maximises P a j b in b M

There exist b oth rigorous and informal arguments in favour of such a deco der Call

a numb er R a reliable transmission rate TR if it is p ossible to vary M and

log M

called the transmission rate of co de M n so that i M n ii the ratio

n

h i

P

remains constant and equals R and iii the errorprobability max P a j b

bM

aa b

vanishes in the limit Call the supremum C sup R R a reliable TR the channel

capacity Then by denition the errornegligible transmission is p ossible for R C and

not for R C How may one assess C or at least check that C The answer was

given by Shannon in we state it informally following the original pap er

There exist a lower b ound for C direct Shannons theorem and an upp er b ound

converse Shannons theorem in terms of an asymptotical b ehaviour of P j as

n Under natural assumptions on P j eg for a memoryless channel the

lower and upp er b ounds coincide and may b e calculated in terms of w j and for

R C the errorprobability decays exp onentially

As noted Shannons result pro duced a strong impression at the time in particular

the existence and a deep physical signicance of a critical value C In his initial

pap ers in Information Theory Dobrushin studied the p ossibility of extending Shannons

theorems to a more general setup where the co ding alphab et is arbitrary He invented

the ab ove condition of information stability which turns out to b e sucient for Shannons

theorems to hold He later extended the theory to the case where the channel statistic

is not known Here is the result of b

Assume that the channel used for transmission is memoryless but the symb olto

symb ol transition probability w j is not given eg may vary from time to time

one knows only that it b elongs to a certain class W The input symb ols b are from an

i

alphab et X and the output ones from Y b oth X and Y are supp osed to b e nite in

the ab ove setup X Y f g Given a probability distribution p p on X and

X

w W set

P

X

pxw x y log w x y

xX

P

I p w

pz W z y

z X

y Y

C C W min I p w

p p

w W

where W is the closed convex hull of W and

C C W max C

p

p

n

Supp ose that at the input of the channel one has words a a a from X the

n

total numb er of the input messages as b efore equals M A co de more precisely a pair

co derdeco der is dened as a set

i

A fa B i M g

i

i n n i j

where a X B Y and a a B B i j M If one receives a

i i j

i

word b B it is deco ded as a The average errorprobability given that the channel

i

statistic while transmitting the sth digit is determined by w W s n equals

s

M n

X X Y

i

w a y w w w eA w

s s n

s

M

s

i

y y B

1 n i

The average errorprobability of co de A for a given class W is

eA sup eA w

n

w W

The b theorem states

For any probability distribution p on X satisfying some additional condition for all

R C W there exists such that for and all n large enough there exists

p

nR

a co de A of length n and size M and with eA Conversely for any

any b etween and and all n large enough there is no co de of length n and size

nC W

with eA

We do not go into detail of the additional condition on p This condition is not

just technical b gives examples where b oth the condition and the assertion of the

theorem fail However this condition apparently excludes only some degenerate variety

of cases

The ab ove results are only a part of Dobrushins activity in Information Theory

We briey mention some others

a Dobrushin discovered and used an elegant formula for mutual entropy b

I I I I

Despite its simplicity it has not previously b een noted in the literature

b He studied the capacity and entropy for some classes of channels and sources

c d

c He established the dep endence of the logarithm of the optimal errorprobability as

a function of co de length c b

d He studied sequential deco ding a

Dobrushins last work c which has strong connections with Information Theory

was devoted to the entropy of Gibbs random elds for high temp eratures

Equilibrium statistical mechanics and the theory of random elds The

problem Dobrushin fo cused on at the b eginning of the s was the construction of prob

abilistic mo dels of matter gas or solid exhibiting the phenomenon of phase transitions

An imp ortant role in his studies was played by where a physical argument is de

velop ed showing that the Ising mo del of a lattice ferromagnet should exhibit a phase

transition when the dimension of the lattice is two or higher Notable publications were

developing useful to ols for describing a state of an innite physical system Do

brushin was of course aware of the famous works and and a numb er of later

pap ers eg giving the exact solution of the twodimensional Ising and related

mo dels However he delib erately and somewhat demonstratively avoided the route of

exact solutions b elieving a qualitative theory to b e the true path towards understanding

complicated phenomena This attitude was typical of him in other elds of research

In ab and a Dobrushin pro duced the rst and rather complicated

version of the pro of of the existence of phase transition in the Ising mo del dimension

two and higher Alternative or similar arguments and pro ofs were given in

and In a Dobrushin develop ed a general approach to the concept of phase

transition as the nonuniqueness of a random eld with a given system of conditional

probabilities or a given sp ecication as it was later called In bc and a

he put this problem in the context of various mo dels of Statistical Mechanics and in

particular gave a new pro of short and b eautiful of the existence of phase transitions

in Isingtyp e mo dels It is this pro of rening the original Peierls argument which is

now presented in most textb o oks and reviews to demonstrate the phenomenon of phase

transitions Additional chapters in ab extend this theory Together with pap ers

and see also these pap ers formed the foundation for

further rapid development of the probabilistic approach to the equilibrium Statistical

Mechanics and later on Euclidean Quantum Field Theory See

The main to ol to describ e a statistical mechanics mo del is the ab ovementioned

d

concept of sp ecication Fix a nite set X and consider the X valued functions Z

d

X on a ddimensional cubic lattice Z Any such function is treated as a conguration

d

of spins on Z the value t gives a state of the spin assigned to the lattice site

d d

t Z Similar denitions hold when Z is replaced by its subset In the Ising mo del

X f g the value t corresp onds to the and t to the

d d

direction of a spin at t Z The space of the congurations on Z is the Cartesian

d

Z

it is endowed with a natural metric and one considers the probability pro duct X

measures on the corresp onding Borel algebra B Any such probability measure P is a

d

random eld RF on Z according to Kolmogorovs theorem it is given by a consistent

d

family of nitedimensional probability distributions P on X where Z is an

arbitrary nite set of lattice sites for many purp oses it is enough to consider lattice

d

cub es fN N N N g

A sp ecication is dened as a consistent family of conditional probability distribu

c d

c c

on the complement Z n on X given a conguration tions P j

called a b oundary condition for A natural physical way to dene such a family is

to a x a collection of functions

X R

c

c

X by b dene functions of two variables X and

X

c

H j

e e e

n

e e

and c set

c c

exp H j P j

c

Z

d

e

As b efore and runs here over nite subsets of Z in is a

e e

n

e

conguration on formed by joining two congurations which is the restriction

e

e e

c

of to and to n the latter is which in turn is the restriction of

e

e

empty when in which case is simply the restriction of to

e e e

n

e

c

in is a normalising constant called the partition function The quantity Z

c

with b oundary condition

X

c c

exp H e j Z



e X



All series in are supp osed to converge absolutely

Pictorially f g is a family of multib o dy interaction p otentials contributing

L

c c

to a conditional energy H j in given b oundary condition the sum in

describ es the energy of interaction of spins t t constituting plus the energy

c

c

An imp ortant observation of their interaction with spins t t constituting

c

is that the distribution P j favours the congurations with a minimal energy

c

H j in physical terminology they are called ground states In the Ising mo del

d

is nonzero only when is a one or a nearestneighb our twop oint subset of Z

t t if jj and ftg

J t t if jj ft t g and kt t k

otherwise

The constant is the inverse temp erature of the system the abundance of

constants is used to analyse in detail the b ehaviour of the mo del in various physical

regions The rst line in describ es a oneb o dy or self interaction physically

it corresp onds to an external magnetic eld and the value t R measures the

d

strength of this eld at site t Z sometimes called the chemical p otential at site

t When t the eld favours the and when t the spin at site t

The second line describ es a twob o dy or binary interaction or coupling of the pairs of

nearestneighb our spins k k stands for the Euclidean norm and J R is called the

coupling constant When J the interaction favours the pairs of neighb our spins

to b e of the same sign in which case the mo del is called ferromagnetic or attractive

whereas J favours them to b e of the opp osite signs in which case the mo del is called

antiferromagnetic or repulsive

Continuing further with ground states GS one can sp eculate that in the ferro

magnetic case with a spacehomogeneous magnetic eld ie t the GS on the

d

whole Z is when and when thus in the absence of the

magnetic eld when one has two comp eting GSs See Fig On the other

d

hand in the antiferromagnetic case with an alternating eld t t if t t Z

are nearestneighb our the GS is a chessb oard conguration where t has the same

sign as t again the case t leads to two GSs See Fig Furthermore in

b oth cases in the absence of the magnetic eld the comp eting GSs exhibit a symmetry

they are transformed into each other by ipping the values of the spin ie changing

them to their opp osites at each lattice site Finally in the ferromagnetic mo del they

are translationinvariant whereas in the antiferromagnetic one translationp erio dic and

transformed into each other by a unit space shift In fact the ab ove congurations

do not exhaust the set of suitably dened GSs but they suce for our immediate

purp oses

Dobrushins tour de force was to treat a sp ecication as a primary ob ject rather than

the random eld The question is how many RFs are there with a given sp ecication

Or if the sp ecication was constructed as in how many RFs do corresp ond to

a given p otential f g He termed them Gibbs random elds with a given p otential

in the case of the Ising mo del one sp eaks of a Gibbs RF with given values of and

J An alternative term used in the literature is Gibbs states we prefer in this article

to follow Dobrushins original terminology The uniqueness of a Gibbs RF is treated

as the absence of a phase transition whereas the nonuniqueness as its presence in this

case it is desirable to describ e the structure of the set of RFs with a given sp ecication

reduced to a study of extremal or at least translationinvariant extremal elements of

this set

Figure

Figure

In certain situations eg when the singlespin space X is innite the problem of the

existence of a Gibbs RF also b ecomes nontrivial

The picture of the structure of the set of Gibbs RFs drawn against the values of

main parameters eg in the case of the ferromagnetic Ising mo del and is called

the phase diagram

At the same time the pap er moved in a similar direction this explains the term

DobrushinLanfordRuelle DLR state or DLR measure

As noted Dobrushin pro duced in bc a concise pro of of the following funda

mental fact

In the Ising mo del with zero magnetic eld ie t the Gibbs RF is unique

when is small ie the temp erature is high and nonunique that is there

are at least two such RFs when is large ie the temp erature is low

Furthermore for and J one of these Gibbs RFs is close to the GS

and another to Similarly for and J one Gibbs RF is close to

the chessb oard GS with O and another to that with O Like the

corresp onding GSs these RFs are translationinvariant in the ferromagnetic mo del and

translationp erio dic in the antiferromagnetic one and are transformed into each other by

ipping the spins and the unit space shift in the antiferromagnetic case as

they converge to the degenerate probability measures sitting exactly at the corresp onding

GSs On the other hand as the unique Gibbs RF converges to the Bernoulli

d

RF with the I ID values t t Z taking values with probability

The pro of was based on an ingenious use of the socalled contour technique going

back to The gap b etween and remains wide it is b elieved that there exists

a critical value separating the uniqueness and nonuniqueness

cr cr

regions In his pap ers Dobrushin also gives a general sucient condition of existence

and uniqueness of a Gibbs RF for a general X We describ e Dobrushins uniqueness

condition for the case when X is nite

d

Let t t Z b e nonnegative numb ers such that

X

t

t

Then if the inequality for the onesite conditional probabilities

X

ftg ftg

t xj d P t xj P

d

Z nftg

Z nftg

xX

X

u t u u

d

uZ u t

d d

holds true for all t Z and all congurations and on Z n ftg there

d

d

Z nftg

Z nftg

exists a unique Gibbs RF with the sp ecication fP g

Using this result he was able to check uniqueness in a large variety of cases including

the ferromagnetic Ising mo del with t and antiferromagnetic with t

t jjt t jj where the GS is unique He also established uniqueness in a

wide class of onedimensional mo dels with d in b and ab

Dobrushin returned to uniqueness problems in a a ab and

a In particular in ab and c a constructive uniqueness criterion was

given which required the verication of a nite though p ossibly large numb er of re

lations Under this criterion the Gibbs RF had many nice prop erties which were

deemed a complete analyticity Here Dobrushins earlier idea of contractivity played

a crucial role

In the nonuniqueness direction Dobrushin following extended the concept of

a contour and stated in explicit form the socalled Peierls condition c This pap er

was an imp ortant step towards the PirogovSinai theory see also

which gave a p owerful metho d of studying the lowtemp erature phase diagrams in the

absence of symmetry

One of the most impressive results achieved by Dobrushin in his study of the non

uniqueness was the theorem for the existence of nontranslationinvariant Gibbs RFs

for the Ising mo del with zero magnetic eld in dimensions three and ab ove The p oint

is that say in the ferromagnetic case the conguration given by

d d d

t if t t t Z and t

d d d

if t t t Z and t

see Fig is also a GS according to a reasonable formal denition The question

is whether the conguration in generates a Gibbs RF in a way similar to the

ab ove translationinvariant GSs In a Dobrushin gave a p ositive answer for d

and large enough Furthermore as he b elieved his results suggest

such that the that in dimension d there should b e another critical value

cr

cr

For but not for nontranslationinvariant Gibbs RFs app ear for

cr

cr cr

the twodimensional Ising mo del d the answer to the ab ove question is negative

and for all there are only two extremal Gibbs RFs Dobrushin came to a

cr

similar conclusion assuming that large enough but in a wider class of twodimensional

mo dels e

Figure

Apart from the ab ove series of results Dobrushin progressed in a variety of other

directions in equilibrium Statistical Mechanics and related areas of Probability Theory

We give b elow a far from exhaustive list of his activities

He analysed logarithmic asymptotics of the partition function and their relation to

the phase diagram a a d He also contributed to the study of the

problem of equivalence of ensembles a

He studied mo dels with a continuous spin space X pro ducing sp ectacular results

b oth for the absence of phase transitions a b and their presence b

a He also considered mo dels with nonstandard interaction p otentials bc

showing unusual features of phase transitions

Investigating into the b ehaviour at a critical p oint he shap ed the theory of auto

mo del selfsimilar RFs in particular he found a new class of Gaussian automo del or

selfsimilar RFs a a c a

In connection with the Euclidean Quantum Field Theory he established various

prop erties of Gaussian RFs be a cd We also mention an attempt

to study the Euclidean phase diagram of the twodimensional b oson P mo del This

attempt was unfortunately not completed after an announcement c there was no

detailed pro of published and the credit justiably was transferred to and

see also but the bypro ducts of these studies lled the pap ers c a

d and provided a score of new results from the theory of generalised Markov RFs

d

on R

A striking example of Dobrushins creative ideas is connected with the problem of

describing sp ecications in terms of a p otential The question is whether the system

conditional probabilities of a Markov or approximately Markov RF can b e written in

a Gibbsian form for a suitable p otential In a sense this may b e regarded as

an inverse problem to problem of phase transition The answer is yes at least under

natural regularity conditions on the sp ecication See and

Dobrushin himself did not publish any result in this direction but his ideas were

used in most of the related publications

In the late s Dobrushin b egan following earlier works a detailed study of

the geometry of random shap es that separate dierent phases in the plane Ising mo del

Heuristically the way of describing such a shap e was discussed in the last century by

the Russian physicist Wul who prop osed a rather general theory of shap es of

surface tension The rigorous form of his theory for the twodimensional Ising mo del

was provided in the monumental pap ers b ab continued in d c

ab and b The extension of this approach to the Ising mo del in higher

dimensions remains an op en challenging problem

As a bypro duct of his studies in the Wul theory Dobrushin b ecame in the late

s deeply interested in the large deviations approach to the theory of phase transitions

c cd Dobrushins idea was that the famous Cramer transformation playing

the crucial role in the analysis of the large deviation probabilities is closely connected to

the Gibbs representation In fact he b elieved in a universality of the large deviations

technique and tried to use it in various areas including Queueing Network Theory see

b elow

Nonequilibrium Statistical Mechanics and pro cesses with lo cal interac

tion At the end of the s Dobrushin showed considerable interest in problems arising

in the theory of random automata networks Initial imp etus for research in this direc

tion was given in the s by PyateckiiShapiro and his coworkers who actively discussed

related problems at a seminar series at MekhMat In Dobrushins interpretation with

d x

each site x of a lattice Z one asso ciates a random pro cess with discrete or continuous

x x

time and the conditional probability sjfs g for pro cess to b e at time t in

y

y d

state s given the states s of pro cesses y Z at the preceding time is determined

y

by the s s with jjy xjj R where R is a constant In the continuoustime setup

y

x

one has in mind the conditional rate of the jump of from s to s Furthermore the

x

x

dierent pro cesses evolve conditionally indep endently One can say that the whole

x d

family f x Z g forms a Markov pro cess with a continual statespace such pro

cesses were later called Markov pro cesses with lo cal interaction In ab Dobrushin

gave a formal construction of such a pro cess and established sucient conditions for

convergence to an invariant distribution He also showed that the reversible invariant

d

distributions are precisely Gibbs RFs on Z with a p otential that is naturally calculated

x

in terms of the conditional probabilities f g His pap ers are considered the origin of a

theory that later b ecame a wellestablished eld of probabilistic research see

Another direction of Dobrushins interests was the construction of the socalled

nonequilibrium dynamical systems of Statistical Mechanics The problem is as follows

Consider a Hamiltonian system of equations in the ddimensional Euclidean space

X

d d

q t p t p t grad V kq t q tk

j j j j k

dt dt

k k j

with the Hamiltonian

X X

A

H fq p g p V kq q k

j j j k

j

j

k k j

d

Here q p R are the p ositions and momenta of particles and V the pair interaction

j j

p otential dep ending on the Euclidean distance the mass of a particle is taken to b e one

A typical shap e of V r r is given in Fig b elow the socalled LennardJones

typ e p otential The value a is the hard core diameter the b ehaviour of V for r a

reects the repulsion when particles are near each other while decreasing as r the

decay of the interaction at large distances For r a V r is supp osed to b e smo oth in

r

Figure

One is interested in solving the Cauchy problem for with an initial date ID

q q p p

j j

j j

If the numb er of particles is nite the solution of exists and is unique for a

massive set of ID fq p g exceptional cases are where jjq q jj a for some j k

j j j

k

and dep ending on the shap e of V for r a other singular initial data eg leading

to triple collisions See eg A similar assertion also holds where the particles

d

are conned to a b ounded domain D R with the b oundary condition of elastic

d

reection or put on the ddimensional torus T However in Statistical Mechanics one

considers the limit as the numb er of particles grows to innity or even deals from the

very b eginning with an innite system of typ e Hamiltonian then b ecomes

as a rule a formal expression and the traditional way of proving the existence and

uniqueness of the solution of the Cauchy problem fails In fact it is p ossible

to construct examples of ID for which the solution blows up or is not unique

However the following remark saves the day for large systems one is interested

not in the evolution of an individual ID but rather of a probability distribution That

is one is concerned with having a solution to not for any ID but for the set

of IDs supp orting a natural probability distribution

First results in this direction were obtained in and for a dierent class of

systems in In particular in the onedimensional case d was considered

with a p otential V that was of a nite range V r for r R and without hard

core ie with no singularity for r The result was that the existence of

and uniqueness for an innite system hold for a massive set of IDs which have

probability one with resp ect to a large class of measures on the phase space of innitely

many particles This set was describ ed in asymptotical terms as well as the class within

which existence and uniqueness held

The ab ove restrictions on d and V were not considered to b e natural and many

researchers tried to remove them An alternative approach was prop osed in and

extended in a series of pap ers completed with here dimension d was ultimately

made arbitrary and the condition on V allowed to include the p otential of the typ e in

Fig However the price to pay was that the set of IDs was made less massive one

could only guarantee that it had probability one with resp ect to any Gibbs RF with

p otential V Such a random eld is dened and constructed in a similar fashion to

the lattice case discussed in Section it turns out to b e invariant or equilibrium

probability distribution under the shift along the solutions of Owing to this fact

results of this kind were referred to as equilibrium dynamical systems In bc

Dobrushin prop osed a new construction of the solution to which allowed him

to include in dimensions d and realistic p otentials V and establish the existence

and uniqueness for a set if IDs having probability one with resp ect to a large class

of measures not necessarily equilibrium ones Up to now these results have not yet

b een improved up on in particular Dobrushin conjectured that in dimension d the

problem of nding a go o d set of IDs has a negative answer

After constructing a dynamical system of innitely many particles one naturally

asks whether it has ergo dic prop erties of one kind or another Dobrushin c

f b b elieved that typically such systems should exhibit convergence to

a limiting distribution at large times and the limit has to b e a Gibbs RF with the

p otential V guring in the original system He even pro duced a physical picture

of such convergence Formally however he was able to check this fact only for some

degenerate mo dels d We give here the corresp onding result for the socalled

onedimensional system of hardro ds Equations of motion may b e formally written in

the form with d and the p otential V r taking values and dep ending on

whether r a or r a where a is the diameter of a hard ro d See Fig b elow

Pictorially the particles move on line R freely when they are apart ie q q

j j

a when they collide ie q q a they exchange their momenta Such a system

j j

may b e considered completely integrable the numb er or fraction of particles with a

given momentum is preserved in time Dealing with hard ro ds it is convenient to think

of a contraction that reduces a hard ro d to a p oint particle the motion of the ro ds is

then transformed into free motion Conversely a dilation map transforms free motion

into that of hard ro ds

An equilibrium invariant distribution P for an innite hardro d system is a random

marked p oint pro cess on the line R with marks momenta from R determined by the

following conditions i the distribution of the p ositions fq g is translationinvariant

j

and given that a p oint is covered by a hard ro d ie jq j a for some i the

i

distances q q a b etween subsequent pairs of ro ds are conditionally I ID and

j j

have an exp onential distribution of mean ii the momenta of the particles are I ID

random variables their common distribution is denoted by The particle density under

The ab ove contraction and dilation maps take this such distribution equals

a

distribution to a Poisson marked pro cess of rate with I ID marks and vice versa

Figure

As was proved in and the equilibrium dynamical system with an invariant

measure of the ab ove typ e has dep ending on go o d ergo dic prop erties Dobrushin

extended such a picture to a wider class of noninvariant measures Namely

Supp ose Q is an arbitrary translationinvariant marked p oint pro cess of density

and with an individual momentum distribution Assume Q satises a condition of

space mixing see d Then the pro cess Q obtained from Q in the course of the

t

hardro d dynamics converges as t to the equilibrium distributions P having the

same density and momentum distribution

The pro of of this result was based on the aforementioned connection with the free

motion for which a similar convergence was established in d e in a wide

situation including any dimension d As noted in Section the main ideas here

go back to e

Continuing further in this direction Dobrushin pioneered the study of the socalled

hydro dynamical limit HL This idea go es back to an earlier pap er in the physical

literature similar attempts may b e traced to the and s see eg and ref

erences therein The problem is to establish a formal connection b etween system

describing the motion of particles ie dynamics at the microlevel and hydro dynami

cal equations Euler and NavierStokes describing the collective motion of a uid or

dense gas medium ie a dynamics at a macrolevel The existing ways of deriving

Euler and NavierStokes equations are essentially heuristic to various degrees the same

is true of other socalled kinetic equations except for the Vlasov equation which Do

brushin exhaustively studied in c For details see The attempt to derive the

Euler equation in was made under certain formal assumptions ab out the solutions

of the socalled BBGKY hierarchy equations A careful analysis of these assumptions

shows that they are related to delicate ergo dic prop erties of system which up to

now have not b een veried and may not hold Dobrushin adopted a dierent p oint of

view he attempted to p erform the HL for sp ecial mo dels where assumptions from

or their equivalent may b e veried and the HL p erformed in a formally correct fashion

These mo dels though nontrivial may b e to o idealistic to lead to the usual Euler or

NavierStokes equations Dobrushin used the term caricatures of hydro dynamics but

they display features of the mechanism b ehind the HL which are b elieved to hold in

more realistic systems Dobrushins results for such caricature mo dels were published

in bd ab b f c e a b a and

b

It has to b e said that Dobrushins pap ers bd ab and b were the

rst ones to contain together the mathematically correct denition and the rigorous

pro of of the HL in the form that is commonly used since then in the mo dern literature

The denition of the HL was indep endently prop osed in on the basis of physical

considerations The main feature of the HL is the spacetime scaling In physical

terms one considers a family of probability distributions fP g of a particle systems

which changes in space on the scale This means that the average parameters of

d

interest calculated around the space p oint q x under P are nice functions of x R

In such parameters were the particle density x the density of momentum px

p x p x and the density of energy ex One then p erforms the shift of P along

d

the solutions of by time t R and calculates in the shifted distribution

the ab ove quantities obtaining functions x p x p x p x P

d 1

and e x more precisely these functions arise as the limits as Under the

assumptions that have b een made in these functions satisfy the Euler equation for

a compressible uid

x divp

p x divp p x P x i d

i i

x

i

e x divep P p x

Parameters x and are related here to the macro whereas q and t are related to the

microscale The quantity P is a function of p and e giving the pressure of the system

with interaction p otential V from It is related to the logarithmic asymptotics of

the partition function with given values of the particle numb er momentum and energy

densities The app earance of the functions p and e is not o ccasional these functions

give the spacetime densities of the fundamental conserved quantities of motion the

numb er of particles the total momentum and the total energy As was shown in

and for a generic p otential V the ab ove canonical rst integrals are the only

p ossible invariants of the motion of an innite system which satisfy a natural additivity

condition it is this condition that allows one to use them in equation and alike

On the other hand there exist exceptional p otentials for which the family of additive

invariants of the motion includes exotic rst integrals In dimension one d these

p otentials have b een investigated in the hardro d p otential on Fig is one of them

Corresp ondingly the onedimensional hardro d system was one of the rst caricature

mo dels to b e investigated in connection with the HL See cd and b As

already observed in this mo del there exists an abundance of the constants of motion

Instead of a triple x p x e x one has to deal here with density x v

of the particles with momentum v at macro p oint x at macro time The ab ove

scheme can then b e carried through and the following quasilinear hyp erb olic partial

dierential equation emerges in the HL

x v

Z Z

v dv v v x v a dw x w x v

x

R

If the initial function ob eys sup dw x w a and sup

1 1

xR xR

R

dw jw j x w then the solution of exists is unique for all R and

satises the same conditions

Equation may b e considered as an analogue of the Euler equation for a hard

ro d uid The hardro d mo del remains the only example of a nonlinear Hamiltonian

system with interaction where the HL was p erformed at a rigorous level with no addi

tional assumption Recently the standard Euler equation was derived in in the

situation where equations of motion include sto chastic terms which remove the main

diculties one had to contend with in

Dobrushin also sp ent a considerable time in thinking of how the NavierStokes

equation should b e related to the HL pro cedure his p oint of view was that it arises

when one takes into account the next correction to the limiting Euler equation up to

the order Such an approach was not unanimously approved among the sp ecialists but

conrmed on caricature mo dels b e and b

A separate although close direction is the HL for various sto chastic mo dels in

cluding pro cesses with lo cal interactions Dobrushins ideas inspired many works in this

eld his own results in this direction are published in b and a

Queueing Network Theory The last eld of Dobrushins research on which we

are going to comment is Queueing Networks QN Theory As was said ab ove he was

driven by fruitful analogies b etween this theory and several areas of Statistical Physics

In b oth elds one deals with a large system characterised by complex interaction b etween

its comp onents It must b e noted that at the b eginning of the s when he b egan work

ing in this direction Queueing Theory was essentially oriented towards problems related

to an isolated device with one or several channels of service Under certain indep endence

and sometimes exp onentiality assumptions an array of elegant formulas was pro duced

for the distribution and exp ected values of various random variables characterising such

a queue waiting time queue size numb er of customers in the queue duration of a

busy or idle p erio d etc Below we use the term the PollyachekKhintchintyp e formulas

when referring to results of this kind see eg

In QN Theory there existed pap ers forming a particular approach to QN

problems the term Jacksons networks was coined for the network class considered in

these pap ers The results of demonstrated striking features of the coming theory

but the consensus was that in general QNs are to o complicated to b e successfully studied

at a mathematically rigorous level The class of Jacksons networks was later considerably

extended but even the new class afterwards called Kellys networks was quite

restrictive for many applications

Dobrushin was not deterred by the apparent complexity of QNs On the other hand

he was not happy with the rather sp ecial exp onentiality and indep endence assumptions

made in the ab ove pap ers leading to the socalled pro ductform of the invariant distribu

tion in Jacksons and Kellys networks This was p erhaps partly b ecause of his general

reservations ab out exactly solvable mo dels First he prop osed the socalled meaneld

approach to the QN theory which he demonstrated in b on the example of a so

called starshap ed messageswitched network

Such a network consists of a centre C and a numb er of input and destination no des

senders and receivers resp ectively S S and D D connected by the di

M N

rected lines as on Fig b elow

Figure

On the input p ort of each line S C and C D there is a single server that

i j

pro cesses or transmits along the line on the FCFS basis a corresp onding stream of

arriving messages Messages originally app ear in no des S S or at the input

M

p orts of lines S C i M one assumes for deniteness that their arrival there

i

is describ ed by I ID Poisson marked pro cesses of rate each with I ID marks

M

The marks in each pro cess are triples b l l where b N is the address of

i

a message and l l the pair of its service or transmission times along lines S C

i

and C D resp ectively The individual mark distribution is as follows a the bs

b

are equidistributed Pb j N b the pair l l is indep endent of b and has

a xed joint distribution eg l and l may b e indep endent or coincide l l

After b eing pro cessed along the line S C a message from with address b j

i i

immediately joins the queue for the line C D After b eing pro cessed along this line

j

it is considered delivered to its destination no de D and disapp ears from the network

j

One is interested in the distribution of the endtoend delay of a given message ie the

time from a messages app earance in no de S to its delivery at D This is clearly the

i j

sum w w l l where w and w are messages waiting times for server S C and

i

server C D resp ectively To formally dene the corresp onding random variables we

j

use the socalled Palm distribution where one of the messages is tagged and followed

through its journey along the path S C D Assume that M N is kept xed and

i j

equal to and the nonoverload conditions

R R

holds where dl dl l dl dl l The distribution

of w is then given by a wellknown PollyachekKhinchintyp e formula for example if

l has the exp onential distribution of rate

exp x x Pw x

x

However the joint distribution of w w and l l and even the marginal distribution

of w cannot b e written in a closed form The theorem in a slightly more general form

than that given in b is as follows

Supp ose that as M N Then the limiting distribution of random variable w

is as in replacing with Furthermore random variables w and w b ecome

asymptotically indep endent and indep endent of the pair l l Hence the limiting

distribution of the endtoend delay is the convolution of those of w and w and l l

This result means that the network with large values of M N op erates as a collection

of nearly indep endent servers each of which has to pro cess a stream of tasks close in

distribution to a p oint pro cess of a simple form in the example under consideration to a

Poisson pro cess The analogy with the meaneld picture in Statistical Mechanics is that

each queue in the network b ecomes asymptotically indep endent of the rest However

the inuence of the whole network on a given server is manifested through the form of

an averaged input stream feeding this server The meaneld approach proved to b e

very rewarding and was later develop ed in a numb er of works See eg the reviews

and a and the references therein as well as the pap er Dobrushin returned to

the meaneldtyp e results in his latest publication a where he studied an example

of a network with elements of control

The network under consideration in a is pictured on Fig b elow It contains

N single servers S S fed with a common exogenous stream of tasks which is

N

assumed to b e Poisson of rate N Let the service times of the tasks b e I ID with the

exp onential distribution of mean and supp ose the nonoverload condition

to b e valid Assume that at the time of tasks arrival one picks randomly a pair of

servers S and S and then selects the one of the two with the shortest queue One

i j

is interested in the distribution of the queue size p er server in such a network It is

clear that this distribution must b e b etter than if the servers were selected completely

randomly the latter mo del can b e solved by means of the PollyachekKhinchintype

formulas It was proved in a that

As N the average queue size distribution in the network with the ab ove selection

rule has the limiting distribution

m

m P Q m

Figure

m

In the case of the completely random choice the probability in equals

and therefore decays much slower with m This illustrates the b enets of the control

intro duced in the mo del

Another direction initiated by Dobrushin in the QN theory was fo cussed on the con

cept of an innite network The aim here was to grasp another feature of the b ehaviour

of complex QNs instability of a stationary regime Formally it should b e manifested in

nonuniqueness of an invariant distribution in a network with innitely many no des as

opp osite to the uniqueness for its nite counterpart This is a striking analogy with the

theory of phase transitions In f Dobrushin has made an initial step in this direc

tion proving that for messageswitched networks on the innite onedimensional lattice

the concept of an invariant distribution may b e correctly dened and in the situation

of small transit ows such a distribution is unique This result inuenced a series of

subsequent pap ers see eg and in particular where Jacksons networks

on innite graphs were analysed and a nonuniqueness of an invariant distribution has

b een established

The third direction in the QN theory connected with the Dobrushins name was con

cerned with general nonoverload conditions guaranteeing the existence of a stationary

regime In analogy with his informationtheoretical studies he used the term the net

work capacity region The problem here is to determine conditions in terms of exp ected

values similar to under which the queues in a given network do not blow up

For Jacksons and Kellys networks these conditions may b e directly derived from the

pro ductform of the invariant distribution Dobrushin b elieved that similar conditions

hold for a general class of networks but his conjecture was later disproved see eg

He has not published any result in this direction but his ideas were instrumental

for a numb er of pap ers see again the reviews and a and a recent work

He also participated in the analysis of the form of an invariant distribution for a general

QN More precisely he asso ciated with a network an inputoutput transformation that

takes an exogenous ow entering the network to the departure ow that leaves it Many

of his predictions ab out the existence of and convergence to an invariant distribution for

such a transformation turned out to b e correct after recent works and

As noted in the late s Dobrushin started an active research of a large deviations

approach to various problems in particular in QN Theory His results are contained in

be In particular in b he analysed the probability of a large deviation for the

waiting time in a tandem singleserver network He discovered the socalled b ottleneck

phenomenon that the logarithmic asymptotics of this probability is determined by the

slowest server The pro of is based on an elegant representation of the waiting time in

terms of the input ow This allowed him to consider wide classes of exogenous pro cesses

in contrast with the most of the pap ers in the eld where one has to intro duce rather

restrictive exp onentiality assumptions

DOBRUSHINS LIST OF PUBLICATIONS

a On regularity conditions for timehomogeneous Markov pro cesses with a countable

set of p ossible states Russian Uspekhi Matematicheskih Nauk No

a A generalization of the Kolmogorov equations for Markov pro cesses with a nite

numb er of p ossible states Russian Matematicheskii Sbornik No

b A limit theorem for Markov chains with two states Russian Izvestia Akademii Nauk

SSSR Seriya Matematicheskaya No translated in Select Transl

Math Stat and Prob Providence Institute of Math Stat and Amer Math So c

pp

a Conditions of regularity for Markov pro cesses with a nite numb er of p ossible states

Russian Matematicheskii Sbornik No

a A lemma on the limit of a sup erp osition of random functions Russian Uspekhi

Matematicheskih Nauk No

b Two limit theorems for the simplest random walk on a line Russian Uspekhi Matem

aticheskih Nauk No

c Central limit theorem for nonstationary Markov chains Russian Doklady AN SSSR

No

a On conditions for the central limit theorem for nonstationary Markov chains Rus

sian Doklady AN SSSR No

b Central limit theorem for nonstationary Markov chains I Russian Teoriya Veroyat

nostei i ee Primeneniya No translated in Theory of Probability and

Its Applications

c Central limit theorem for nonstationary Markov chains I I Russian Teoriya Veroy

atnostei i ee Primeneniya No translated in Theory of Probability

and Its Applications

d An example of a countable homogeneous Markov pro cess all states of which are

transient Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

e On Poisson laws for the distributions of particles in space Russian Ukrainskii

Mathematicheskii Zhurnal No

f With AM Jaglom A complement of the translators Russian In J Doob Stokhas

ticheskie Protsessy Russian Moscow Izdatelstvo Inostrannoii Literatury pp

a complement to the the Russian translation of Do ob JL Stochastic processes

New York Wiley

a Some classes of homogeneous denumerable Markov pro cesses Russian Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of

Probability and Its Applications

a The continuity condition of a sample function of a martingale Russian Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of Proba

bility and Its Applications

b A statistical problem arising in the theory of detection of signals in the presence

of noise in a multichannel system and leading to stable distribution laws Russian

Teoriya Veroyatnostei i ee Primeneniya No translated in Theory

of Probability and Its Applications

c Information transmission in a channels with feedback Russian Teoriya Veroyatnostei

i ee Primeneniya No translated in Theory of Probability and Its

Applications

d A simplied metho d of exp erimentally evaluating the entropy of a stationary sequence

Russian Teoriya Veroyatnostei i ee Primeneniya No translated

in Theory of Probability and Its Applications

a A general formulation of the basic Shannon theorem in information theory Russian

Doklady AN SSSR No

b A general formulation of the basic Shannon theorem in information theory Russian

Uspekhi Matematicheskih Nauk No translated in AMS Translations

Series

c Optimum information transmission through a channel with unknown parameters

Russian Radiotehnika i Electronika No translated in Radio

Engineering and Electronics No

a Passage to the limit under the information and entropy signs Russian Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of Proba

bility and Its Applications

b Prop erties of sample functions of a stationary Gaussian pro cess Russian Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of

Probability and Its Applications

c The asymptotic b ehavior of the probability of errors when information is transmit

ted through channel without memory with symmetric matrix of transition probabilities

Russian Doklady AN SSSR No translated in Soviet Mathe

matics Doklady

d With YaI Hurgin and BS Tsybakov An approximate computation of the capacity

of radio chanells with random parameters Russian Trudy Vsesoyuznogo Soveschania po

Teorii Veroyatnosteii i Matematicheskoii Statistike Yerevan Yerevan Izdatelstvo

AN Arm SSR pp

e With AM Jaglom and IM Jaglom Theory of information and linguistics Rus

sian Voprosy Yazykoznaniya No

a Mathematical metho d in linguistics Russian Matematicheskoe Prosveschenie

Moscow Fizmatgiz pp

b Mathematical problems in the Shannon theory of optimal co ding of information

Russian Problemy Peredachi Informatsii Moscow Izdatelstvo AN SSSR

pp translated in Proc Fourth Berkeley Symp on Math Stat and Prob

Berkeley University of California Press pp

a Optimal binary co des for small rates of transmission of information Russian Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of

Probability and Its Applications

b Asymptotic estimates of the probability of error for transmission of messages over

a discrete memoryless communication channel with a symmetric transition probability

matrix Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

c Asymptotic estimates of the probability of error for the transmission of messages over

a memoryless channel with feedback Russian Problemy Kibernetiki

d With BS Tsybakov Information transmission with additional noise IEEE Trans

Inform Theory

a Asymptotic optimality of group and systematic co des for some channels Russian

Teoriya Veroyatnostei i ee Primeneniya No translated in Theory of

Probability and Its Applications

b Individual metho ds for the transmission of information for discrete channels without

memory and messages with indep endent comp onents Russian Doklady AN SSSR

No translated in Soviet Mathematics Doklady

c Unied metho ds for the transmission of information a general case Russian Doklady

AN SSSR No translated in Soviet Mathematics Doklady

d With MS Pinsker and AN Shiryaev An application of the entropy to problems

of signal detecting against a background of noise Russian Litovskii Matematicheskii

Sbornik No

e With RA Minlos Possibilities of applications of limit theorems of probability the

ory to some physical problems Russian Predelnye Teoremy Teorii Veroyatnosteii

Tashkent FAN

a On the WozencraftReien metho d of sequential deco ding Russian Problemy Kiber

netiki

b Investigation of conditions for the asymptotic existence of the congurational integral

of the Gibbs distribution Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

c Metho ds of probability theory in statistical physics Russian In Trudy Zimnieii

Shkoly po Teorii Veroyatnosteii i Matematicheskoii Statistike Uzhgoro d Kiev

Izdatelstvo KGU pp

a Existence of a phase transition in the twodimensional Ising mo dels Russian Doklady

AN SSSR No translated in Soviet Physics Doklady

b Existence of a phase transition in two and three dimensional Ising mo dels Russian

Teoriya Veroyatnostei i ee Primeneniya No translated in Theory

of Probability and Its Applications

a Existence of phase transitions in mo dels of a lattice gas Proc Fifth Berkeley Symp

on Math Stat and Prob vol Berkeley University of California Press

pp

b Theory of optimal co ding of information Russian Kibernetiku na Sluzhbu Kom

munizmu Moscow Energiya pp

a With RA Minlos Existence and continuity of the pressure in classical statistical

physics Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

b Shannon theorems for channels with synchronization errors Russian Problemy

Peredachi Informatsii No translated in Problems of Information

Transmission No

a The description of a random eld by means of conditional probabilities and conditions

of its regularity Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

b Gibbsian random elds for lattice systems with pairwise interactions Russian Funk

tsionalnyi Analis i ego Prilozheniya No translated in Functional

Analysis and Its Applications

c The problem of uniqueness of a Gibbsian random eld and the problem of phase

transition Russian Funktsionalnyi Analis i ego Prilozheniya No

translated in Functional Analysis and Its Applications

d With N Vvedenska ja Calculation on a computer of the capacity of communication

channels with exclusion of symb ols Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission No

a Gibbsian elds General case Russian Funktsionalnyi Analis i ego Prilozheniya

No translated in Functional Analysis and Its Applications

b With II PjatetskiiShapiro and NB Vasiljev Markov pro cesses on an innite pro d

uct of discrete spaces Russian Proc of the SovietJapanese Symposium in Probability

Theory Khabarovsk June KhabarovskNovosibirsk pp

c With MS Pinsker Memory increases transmission capacity Russian Problemy

Peredachi Informatsii No translated in Problems of Information

Transmission

a Gibbsian random elds for particles without hard core Russian Teoreticheskaya i

Matematicheskaya Fizika No

b Prescribing a system of random variables by conditional distributions Russian

Teoriya Veroyatnostei i ee Primeneniya No translated in Theory

of Probability and Its Applications

c Unied metho ds of optimal quantization of messages Russian Problemy Kibernetiki

a Markov pro cesses with a large numb er of lo cally interacting comp onents existence of

a limit pro cesses and their ergo dicity Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission

b Markov pro cesses with many lo cally interacting comp onents the reversible case and

some generalizations Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission

c With RA Minlos and YuM Suhov A review of some recent results Russian

In D Ruelle Statisticheskaya Mekhanika Strogie Resultaty Moscow Mir pp

a complement to the Russian translation of the b o ok D Ruelle Statistical

Mechanics Rigorous Results New York Benjamin

a Asymptotic b ehaviour of Gibbsian distributions for lattice systems and its dep endence

on a form of a volume Russian Teoreticheskaya i Matematicheskaya Fizika No

translated in Theoretical and Mathematical Physics

b Gibbs states describing a co existence of phases for the threedimensional Ising mo del

Russian Teoriya Veroyatnostei i ee Primeneniya No translated

in Theory of Probability and Its Applications reprinted in Math

Problems of Statistical Mechanics Singap ore World Scientic

With SI Gelfand Complexity of asymptotically optimal co de realization by constant

depth schemes Problems of Control and Information Theory

d Survey of Soviet research in information theory IEEE Trans Inform Theory

a An investigation of Gibbsian states for threedimensional lattice systems Teoriya

Veroyatnostei i ee Primeneniya No translated in Theory of

Probability and Its Applications reprinted in Math Problems of

Statistical Mechanics Singap ore World Scientic

b Analiticity of correlation functions in onedimensional classical systems with slowly

decreasing p otentials Commum Math Phys

c With RA Minlos Construction of onedimensional quantum eld via a continuous

Markov eld Russian Funktsionalnyi Analis i ego Prilozheniya No

translated in Functional Analysis and Its Applications

d Mathematization of linguistics Russian Izvestia Academii Nauk SSSR Ser Liter

atura Linguistika No

e With SI Gelfand and MI Pinsker On complexity of co ding Proc Second Inter

national Symposium on Information Theory Tsahkadzor Armenia Septemb er

Budap est Academia Kiado pp

a Conditions of absence of phase transitions in onedimensional classical systems Rus

sian Matematicheskii Sbornik No translated in Mathematics of

the USSR Sbornik

b Analyticity of correlation functions in onedimensional classical systems with a slowly

p ower decrease of p otential Russian Matematicheskii Sbornik No trans

lated in Mathematics of the USSR Sbornik

c With VM Gertsik Gibbsian states in a lattice mo del with the twostep interaction

Russian Funktsionalnyi Analis i ego Prilozheniya No translated

in Functional Analysis and Its Applications

d With BS Nakhap etjan Strong convexity of the pressure for lattice systems of

classical statistical physics Russian Teoreticheskaya i Matematicheskaya Fizika

No translated in Theoretical and Mathematical Physics

a With SB Shlosman Absence of breakdown of continuous symmetry in

twodimensional mo dels of statistical physics Commun Math Phys

b With SZ Stambler Co ding theorems for classes of arbitrary varying discrete

memoryless channels Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission

c With RA Minlos Quotient measures on measurable spaces Russian Trudy

Moskovskogo Matematicheskogo Obschchestva translated in Transac

tions of the Moscow Mathematical Society

a With RM Minlos An investigation of prop erties of generalized Gaussian random

elds Russian In Zadachi Mekhaniki i Matematicheskoii Fiziki Moscow Nauka pp

translated in Selecta Mathematica Sovietica

b With YuM Sukhov Asymptotical investigation of starshap e message switching

networks with a large numb er of radial rays Russian Problemy Peredachi Informatsii

No translated in Problems of Information Transmission

c With SA Pirogov Theory of random elds Proc of the IEEE USSR Joint

Workshop on Information Theory New York IEEE Press pp

a With B Tirozzi The central limit theorem and the problem of equivalence of

ensembles Commun Math Phys

b With J Fritz Nonequilibrium dynamics of onedimensional innite particle systems

with a singular interaction Commun Math Phys

c With J Fritz Nonequilibrium dynamics of twodimensional innite particle systems

with a singular interaction Commun Math Phys

d With RA Minlos Polynoms in linear random functions Russian Uspekhi Matem

aticheskih Nauk No translated in Russian Mathematical Surveys

Uspekhi No

e With SI Ortjukov Lower b ound for the redundancy of selfcorrecting arrangements

of unreliable functional elements Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission

f With SI Ortjukov Upp er b ound for the redundancy of selfcorrecting arrangements

of unreliable functional elements Russian Problemy Peredachi Informatsii No

translated in Problems of Information Transmission

a Automo del generalized random elds and their renormgroups Russian In Mnogo

componentnye Sluchaiinye Sistemy Moscow Nauka pp translated in

Multicomponent Random Systems eds RL Dobrushin and YaG Sinai Advances in

Probability and Related Topics vol New York Marcel Dekker pp

b With SB Shlosman Nonexistence of one and twodimensional Gibbs elds with

a noncompact group of continuous symmetries Russian In Mnogocomponentnye

Sluchaiinye Sistemy Moscow Nauka pp translated in Multicompon

ent Random Systems eds RL Dobrushin and YaG Sinai Advances in Probability

and Related Topics vol New York Marcel Dekker pp

c With YuM Sukhov On the problem of the mathematical foundation of the Gibbs

p ostulate in classical statistical mechanics In Mathematical Problems in Theoretical

Physics eds G DellAntonio S Doplicher and G IonaLasinio Lecture Notes in

Physics Berlin SpringerVerlag pp

d With RA Minlos Polynomials of a generalized random eld and its moments

Russian Teoriya Veroyatnostei i ee Primeneniya No translated

in Theory of Probability and Its Applications

a Gaussian and their sub ordinated selfsimilar random generalized elds Ann Prob

b With D Surgailis On the innovation problem for Gaussian Markov random elds

Z Wahrsch Verw Geb

c The Vlasov equation Russian Funktsionalnyi Analis i ego Prilozheniya No

translated in Functional Analysis and Its Applications

d With YuM Sukhov Time asymptotics for some degenerate mo dels of time evo

lution for system with innitely many particles Russian In Itogi Nauki i Tekhniki

Ser Sovremennye Problemy Matematiki Moscow VINITY pp

translated in J Soviet Mathematics

e With P Ma jor Noncentral limit theorems for nonlinear functions of Gaussian

elds Z Wahrsch Verw Geb

f With VV Prelov Asymptotic approach to the investigation of message switch

ing networks of a linear structure with a large numb er of centers Russian Problemy

Peredachi Informatsii No translated in Problems of Information

Transmission

g With MYa Kelb ert Lo cal additive functionals of Gaussian random elds Rus

sian Uspekhi Matematicheskih Nauk No translated in Russian

Mathematical Surveys Uspekhi No

a Gaussian Random Fields Gibbsian p oint of view Russian In Multicomponent

Random Systems eds RL Dobrushin and YaG Sinai Advances in Probability and

Related Topics vol New York Marcel Dekker pp

b With YaG Sinai Mathematical problems in statistical mechanics In Soviet Sci

entic Reviews Section C Mathematical Physics Reviews Chur Harwo o d Academic

Publishers pp

c With C Boldrighini and YuM Suhov Hydro dynamical limit for a degenerate

mo del of classical statistical mechanics Russian Uspekhi Matematicheskih Nauk

No

d With C Boldrighini and YuM Suhov Hydro dynamics of onedimensional hard

ro ds Russian Uspekhi Matematicheskih Nauk No

e With C Boldrighini and YuM Suhov Time Asymptotics for Some Degenerate

Models of Time Evolution for System with Innitely Many Particles Camerino Uni

versity Publishing House

a With EA Pechersky Uniqueness conditions for nitely dep endent random elds

In Random Fields eds J Fritz JL Leb owitz and D Szasz vol Amsterdam

NorthHolland pp

b With SB Shlosman Phases corresp onding to minima of the lo cal energy Selecta

Mathematica Sovietica

c With P Ma jor On asymptotical b ehavior of some selfsimilar random elds Selecta

Mathematica Sovietica

d With RM Gray and DS Ornstein Blo ck synchronization slidingblo ck co ding

invulnerable sources and zero error co des for discrete noisy channels Ann of Prob

a With R SiegmundSchultze The hydro dynamic limit for systems of particles with

indep endent evolution Math Nachr

b Hydro dynamic limit approach some caricatures Russian In Vzaimodeiistvuy

uschchie Markovskie Protsessy i ih Primeneniya k Matematicheskomu Modelirovaniyu

Biologicheskih Sistem Puschchino Moscow region pp

a With EA Pechersky A criterion of the uniqueness of Gibbsian elds in the non

compact case In Probability Theory and Mathematical Statistics Proceedings of the

Fourth USSRJapan Symposium eds K Ito and JV Prokhorov Lecture Notes in

Mathematics Berlin SpringerVerlag

b With C Boldrighini and YuM Suhov One dimensional hard ro d caricature of

hydro dynamics J Stat Phys

c With MYa Kelb ert Lo cal additive functionals of Gaussian random elds Rus

sian Teoriya Veroyatnostei i ee Primeneniya No translated in

Theory of Probability and Its Applications

d With MYa Kelb ert Stationary lo cal additive functionals of Gaussian elds

Russian Teoriya Veroyatnostei i ee Primeneniya No translated

in Theory of Probability and Its Applications

a With SB Shlosman Constructive criterion for the uniqueness of Gibbs eld In

Statistical Physics and Dynamical Systems Rigorous results Progress in Physics vol

Boston Birkhauser pp

b With SB Shlosman Completely analytical Gibbs elds In Statistical Physics and

Dynamical Systems Rigorous Results eds J Frtitz A Jae and D Szasz Progress

in Physics vol Boston Birkhauser pp

c With I Kolafa and SB Shlosman Phase diagram of the two dimensional Ising

antiferromagnet Commun Math Phys

d With MG Avetisjan A condition of the linear regularity for vector random elds

Russian Problemy Peredachi Informatsii No translated in Prob

lems of Information Transmission No

e With SB Shlosman The problem of translation invariance of Gibbs states at low

temp erature In Soviet Scientic Reviews Section C Mathematical Physics Reviews

Chur Harwo o d Academic Publishers pp

f With YaG Sinai and YuM Sukhov Dynamical systems of statistical mechanics

Russian In Itogi Nauki i Tekhniki Ser Sovremennye Problemy Matematiki Funda

mentalnye Napravleniya Vol Dinamicheskie Systemy Moscow VINITI pp

translated in Dynamical Systems II Encyclopaedia of Mathematical Sciences

vol ed YaG Sinai Berlin SpringerVerlag pp

a With M Zahradnik Phase diagrams for continuousspin mo dels An extension

of the PirogovSinai theory In Mathematical Problems of Statistical Mechanics and

Dynamics Dordrecht D Reidel pp

b With LA Bassalygo Uniqueness of a Gibbs eld with random p otential an ele

mentary approach Russian Teoriya Veroyatnostei i ee Primeneniya No

translated in Theory of Probability and Its Applications

c With A Pellegrinotti YuM Suhov and L Triolo Onedimensional harmonic lattice

caricature of hydro dynamics J Stat Phys

a With SB Shlosman Completely analytical interactions constructive description

J Stat Phys

b Induction on volume and cluster expansion In Proc VIII International Congress on

Mathematical Physics eds M Mebkhout and B Seneor Singap ore World Scientic

pp

c With LA Bassalygo entropy of Gibbsian elds Russian Problemy Peredachi

Informatsii No translated in Problems of Information Transmission

d Theory of information Russian In AN Kolmogorov Izbrannyie Trudy Teoriya

Informatsii i Teoriya Algoritmov Moscow Nauka pp

e Switching networks Gibbsian elds interconnections In Proc st World Congress

of the Bernoul li Society Tashkent Utrecht VN Sci Press pp

a A new approach to the analysis of Gibbs p erturbations of Gaussian elds Selecta

Mathematica Sovietica

b With MR Martirosjan Nonnite p erturbation of Gibbsian elds Russian Teo

reticheskaya i Matematicheskaya Fizika No translated in Theoret

ical and Mathematical Physics

c With MR Martirosjan A p ossibility of hightemp erature phase transition due

to manyparticle character of p otential Russian Teoreticheskaya i Matematicheskaya

Fizika No translated in Theoretical and Mathematical Physics

d With J Fritz and YuM Suhov AN Kolmogorov the founder of the theory of

reversible Markov pro cesses Russian Uspekhi Matematicheskih Nauk No

translated in Russian Mathematical Surveys Uspekhi No

e With A Pellegrinotti YuM Suhov and L Triolo Onedimensional harmonic lattice

caricature of hydro dynamics Second approximation JStat Phys

a Caricatures of Hydro dynamics In Proc IX International Congress on Mathematical

Physics eds B Simon A Truman and IM Davies Bristol Adam Higler pp

b With R Kotecky and SB Shlosman Equilibrium crystal shap es a microscopic

pro of of the Wul construction In Proc of XXIV Karpacz Winter School Stochastic

Methods in Mathematical Physics Singap ore World Scientic pp

a With MYa Kelb ert AN Rybko and YuM Suhov Qualitative metho ds of queue

ing network theory In Stochastic Cel lular Systems Ergodicity Memory Morphogen

esis eds RL Dobrushin VM Kryukov and AL To om Manchester University

Press pp

b With A Pellegrinotti and YuM Suhov Onedimensional harmonic lattice carica

ture of hydro dynamics The higher corrections J Stat Phys

c With V Warstat Completely analytic interactions with ininite values Prob The

ory Rel Fields

a With F Sokolovskii Higher order hydro dynamic equations for a system of inde

p endent random walks In Random walks Brownian Motion and Interacting Particle

Systems A Festschrift in Honor of Frank Spitzer eds R Durrett and H Kesten

Boston Birkhauser pp

a With R Kotecky and SB Shlosman Wul construction A Global Shape from

Local Interaction Translations of Mathematical Monographs vol Providence

RI American Mathematical So ciety p

b With SB Shlosman Thermo dynamic inequalities for the surface tension and the

geometry of the Wul construction In Ideas and Methods in Mathematical Analysis

Stochastics and Application In Memory of R HoeghKrohn eds S Alb everio JE

Fenstad H Holden and JE Lindstrom vol Cambridge Cambridge University

Press pp

c With SB Shlosman Large deviations b ehavior of statistical mechanics mo dels in

multiphase regime In Proc X International Congress on Mathematical Physics ed

K Schmugden Berlin SpringerVerlag pp

a A formula of full semiinvariants In Cel lular Automata and Cooperative Systems eds

N Bo ccara EGoles S Martinez PPicco Dordrecht Kluwer pp

b On the way to the mathematical foundations of statistical mechanics In RL

Dobrushin and S Kusuoka Statistical Mechanics and Fractals Lecture Notes in Math

ematics Berlin SpringerVerlag pp

c A statistical b ehaviour of shap es of b oundaries of phases In Phase Transitions

Mathematics Physics Biology ed R Kotecky Singap ore World Scientic

pp

d With R Kotecky and SB Shlosman A microscopic justication of the Wul con

struction J Stat Phys

a Estimates of semiinvariants for the Ising mo del at low temp eratures Preprint No

Vienna ESI

b With EA Pechersky Large deviations for tandem queueing systems Journal of

Applied Mathematics and Stochastic Analysis

c With SB Shlosman Large and mo derate deviations in the Ising mo del In Prob

ability Contributions to Statistical Mechanics ed RLDobrushin Advances in Soviet

Mathematics vol Providence American Mathematical So ciety pp

d With SB Shlosman Droplet condensation in the Ising mo del mo derate deviations

p oint of view In Probability and Phase Transition ed G Grimmett Dordrecht

Kluwer

e With VM Blinovsky Pro cess level large deviations for a class of piecewise ho

mogeneous random walks In The Dynkin Festschrift Markov Processes and Their

Applications ed MI Freidlin Progress in Probability vol Boston Birkhauser

pp

a With OO Hryniv On uctuations of Wulf s shap e in the two dimension Ising

mo del Russian Uspekhi Matematicheskih Nauk No translated

in Russian Mathematical Surveys Uspekhi No

b With OO Hryniv On uctuations of the phase b oundary in the D Ising ferromag

net Preprint No Vienna ESI

a With ND Vvedenskaya and FI Karp elevich A queueing system with selection

of the shortest of two queues an asymptotical approach Problems of Information

Transmission

b With OO Hryniv Fluctuations of shap es of large areas under paths of random

walks Prob Theory Rel Fields

Acknowledgements The authors would like to thank L Bassalygo C Boldrighini G

Dobrushina B Gurevich A Jae R Kotecky L Philipp ova M Pinsker S Shlosman

B Tsybakov and N Vvedenskaya for valuable comments on various matters discussed

in the article The hospitality of the The ErwinSchrodinger Institute and of the or

ganisers of the Dobrushin Memorial Conference Statistical Mechanics as a branch of

the Probability Theory Vienna Septemb er is particularly appreciated

Sp ecial thanks go to Sarah SheaSimonds for correcting the style of the pap er

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