Selected Title s i n Thi s Subserie s

47 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s way . Prom probabilit y theor y t o statistica l physic s (TRANS2/198 ) 46 Vladimi r Arnold , Maxi m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c topology (TRANS2/197 ) 45 Ya . Eliashberg , D . Fuchs , T . Ratiu , an d A . Weinstein , Editors , Norther n California symplecti c geometr y semina r (TRANS2/196 ) 44 Alexande r Astashkevic h an d Serg e Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras, an d Application s (D . B. Fuchs' 60t h Anniversar y Collection) (TRANS2/194 ) 43 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematica l Physics (TRANS2/191 ) 42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knots an d Curve s (TRANS2/190) 41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d Spectral Theor y (M . Sh . Birman's 70t h anniversar y collection ) (TRANS2/189 ) 40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y (TRANS2/187) 39 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l Equations (TRANS2/186 ) 38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e Invariants (Mathematic s a t th e Independen t Universit y o f ) (TRANS2/185 ) 37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s (Dedicated t o Seli m Krein ) (TRANS2/184 ) 36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 ) 35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 ) 34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y Theory (TRANS2/180 ) 33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n the Crossroa d (TRANS2/179 ) 32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/177) 31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 ) 30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Netsvetaev , Editors , Mathematic s in St . Petersbur g (TRANS2/174 ) 29 V . Kharlamov , A . Korchagin , G . Polotovskit , an d O . Viro , Editors , Topolog y o f Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 ) 28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w Seminar o n Dynamica l System s (TRANS2/171 ) 27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 ) 26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras: E . B . Dynkin's Semina r (TRANS2/169 ) 25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 ) 24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l Equations (TRANS2/167 ) 23 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m (TRANS2/165) 22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 ) (Continued in the back of this publication) This page intentionally left blank On Dobrushin's Way . From Probability Theor y to Statistical Physic s

http://dx.doi.org/10.1090/trans2/198

American Mathematical Societ y TRANSLATIONS

Series 2 • Volum e 19 8

Advances in the Mathematical Sciences — 47

On Dobrushin's Way . From Probabilit y Theor y to Statistical Physic s

R. A. Minlos Senya Shlosma n Yu. M . Suhov Editors

P.JBEM47.,,

American Mathematical Societ y Providence, Rhode Islan d ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E V. I . ARNOL D S. G . GINDIKI N V. P . MASLO V

2000 Subject Classification. Primar y 60K35 , 82Bxx .

ABSTRACT. The volume contains research articles in , especially in statistica l mechanics, writte n b y friends , collaborators , an d student s o f Rolan d L/vovic h Dobrushi n (1929 - 1995). Als o include d ar e a shor t biograph y an d som e recollection s abou t Dobrushin . The boo k i s useful fo r researcher s an d graduat e student s workin g i n mathematical physic s an d Markov processe s an d fields .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-2150- 4 ISSN 0065-929 0

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© 200 0 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . © Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 4 3 2 1 0 5 0 4 03 0 2 01 0 0 Contents

Foreword

A brief biograph y

The lowe r spectral branc h o f the generato r o f the stochasti c dynamic s fo r the classica l Heisenber g mode l N. ANGELESCU , R . A . MINLOS , AN D V . A . ZAGREBNO V

Random wal k i n a fluctuating rando m environmen t wit h Marko v evolutio n C. BOLDRIGHINI , R . A . MINLOS , AN D A . PELLEGRINOTT I

Ergodicity an d exponentia l convergenc e o f a Glauber-f-Kawasak i proces s S. BRASSESCO , E . PRESUTTI , V . SIDORAVICIUS , AN D M . E . VARE S

The Griffith s singularit y rando m field A. VA N ENTER , C . MAES , R . H . SCHONMANN , AN D S . SHLOSMA N

Dobrushin's progra m o n Gibbsianit y restoration : Weakl y Gibb s and almos t Gibbs random fields AERNOUT VA N ENTER , CHRISTIA N MAES , AN D SENY A SHLOSMA N

Space-time invarian t state s o f the idea l ga s with finite number , energy , an d entropy densit y GREGORY L . EYIN K AN D HERBER T SPOH N

Hausdorff dimensio n an d pressur e i n the DL R thermodynamic formalis m B. M . GUREVIC H AN D A . A . TEMPELMA N

Nonsymmetric simpl e random walk s along orbits o f ergodic automorphism s V. Yu . KALOSHI N AN D YA . G . SINA I

The Crame r transfor m an d larg e deviation s o n three-dimensiona l Lobachevsky spac e F. I . KARPELEVICH , E . A . PECHERSKY , AN D YU . M . SUHO V

Thermodynamical limi t fo r symmetri c close d queuin g network s F. I . KARPELEVIC H AN D A . N . RYBK O

Random infinit e spi n grap h evolutio n V. A . MALYSHE V viii CONTENT S

An elementary approac h t o finite siz e conditions fo r the exponentia l decay o f covariances i n lattice spi n model s F. MARTINELL I Dynamics o f Ising spin system s a t zer o temperatur e S. NANDA , C . M . NEWMAN , AN D D . L . STEI N Peierls argument fo r the anisotropi c Isin g mode l S. A . PIROGO V Contour method s an d Pirogov-Sina i theor y fo r continuou s spi n lattic e models MILOS ZAHRADNI K Recollections Foreword

1999 mark s th e 70t h birthda y anniversar y o f Rolan d Dobrushin— a brillian t mathematician an d a remarkable person. Fou r years have already gon e by since he passed away , but hi s memory i s still very muc h aliv e and the feelin g o f irreparabl e loss only increase s a s the years g o by. R. Dobrushi n worke d i n severa l branche s o f mathematics (probabilit y theory , ) , bu t hi s deepes t influenc e wa s o n mathematica l physics . H e was on e o f th e founder s o f th e rigorou s stud y o f statistica l physics . I n th e earl y sixties, whe n R . Dobrushi n bega n workin g i n that direction , onl y a fe w peopl e i n the world wer e thinking alon g the same lines , whereas no w there i s a huge army o f researchers i n this field. Th e presen t collectio n o f articles i n mathematical physic s is devoted t o the memor y o f Roland Dobrushin . Th e author s participatin g i n thi s collection kne w Rolan d quit e well ; some were actually hi s collaborators . The title o f the volume , "O n Dobrushin's Way " stresse s the fac t tha t th e cur - rent developmen t o f mathematica l physic s i s indee d evolvin g alon g th e line s tha t Dobrushin foresaw , an d als o that th e ideas and methods due to him are extensivel y used i n this discipline . In addition to research papers, this volume contains a short biography and some recollections abou t R . Dobrushin . Th e latte r ar e organize d i n thre e sections : th e first ar e recollections related to Dobrushin's socia l and political outlook, the secon d are personal reminiscence s o f his contemporaries, the third consist s o f recollection s of his younger colleagues . W e hope that fro m thes e shor t an d somewha t disparat e sketches the reade r wil l be abl e to experienc e a bit o f the char m o f Roland's mar - velous personality . A detaile d accoun t o f R . L . Dobrushin' s researc h activitie s ca n b e foun d i n [l]-[8]-

References [1] L . A. Bassalygo, V. A. Malyshev, R . A. Minlos, I. A. Ovseevich, E. A. Pecherski, M . S. Pinsker , V. V. Prelov, A . N. Rybko, Yu . M. Suhov, and S . B. Shlosman, In memory of Roland V vovich Dobrushin (with list of publications), Problem y Peredach i Informatsi i 3 2 (1996) , no. 3 , 3-24 ; English transl . Problem s Inform . Transmissio n 3 2 (1996) , no . 3 , 223-238. [2] R . A . Minlos , E . A . Pechersky , an d Yu . M . Suhov , Remarks on the life and the research of Roland L. Dobrushin, J . Appl . Math . Stochasti c Anal . 9 (1996) , no. 4 , 337-372 . [3] I . A. Ibragimov, R. L. Dobrushin's work on Markov processes, Uspekh i Mat . Nau k 5 2 (1997) , no. 2 , 5-9; Englis h transl . n Math . Survey s 5 2 (1997) , no. 2 , 239-243. [4] V . V. Prelov, Information theory in R. L. Dobrushin's research, Uspekh i Mat. Nauk 52 (1997) , no. 2 , 9-13; English transl . Russia n Math . Survey s 5 2 (1997) , no. 2 , 245-249 . [5] R . A. Minlos, R. L. Dobrushin —one of the founders of modern mathematical physics, Uspekh i Mat. Nau k 5 2 (1997) , no . 2 , 13-18 ; Englis h transl . Russia n Math . Survey s 5 2 (1997) , no . 2 , 251-256. ix x FOREWOR D

[6] B . M. Gurevich , Dynamical aspects of statistical physics in R. L. Dobrushin's works, Uspekh i Mat. Nau k 5 2 (1997) , no . 2 , 18-25 ; Englis h transl . Russia n Math . Survey s 5 2 (1997) , no . 2 , 257-264. [7] E . A . Pechersky an d Yu . M . Suhov , The ideas of R. L. Dobrushin in queuing theory, Uspekh i Mat. Nau k 5 2 (1997) , no . 2 , 25-30 ; Englis h transl . Russia n Math . Survey s 5 2 (1997) , no . 2 , 265-271. [8] B . M. Gurevich, I . A. Ibragimov, R . A . Minlos, I . A. Ovseevich, V . I. Oseledets, M. S . Pinsker , V. V. Prelov , Yu . V. Prokhorov , Ya . G . Sinai , an d A . S . Kholevo, Roland L'vovich Dobrushin (July 20 , 1929-December 12 , 1995), Teor. Veroyatnost. i Primenen. 41 (1996) , no. 1 , 164-169 ; English transl . Theor y Probab . Appl . 41 (1996) , no. 1 , 132-136 .

Translated b y A . B . SOSSINSK Y A brie f biograph y

R. L . Dobrushi n wa s born o n Jul y 20th , 192 9 in Leningrad . A t th e ag e o f si x he lost hi s father an d move d to Mosco w with hi s mother. Soo n after th e end o f the war hi s mother als o died, an d h e was brought u p i n a famil y o f relatives. R. L . Dobrushin , whos e lov e fo r mathematic s aros e i n hi s hig h schoo l years , entered the Mechanics and Mathematics Department o f Moscow University in 194 7 and graduated fro m i t i n 1952 . Beginnin g fro m hi s freshman year , R . L. Dobrushi n participated i n the optional student semina r heade d by E. B. Dynkin, where he ac- quired his interest i n probability theory as well as the very specific styl e o f thinking characteristic o f expert s i n tha t field. I n subsequen t year s R . L . Dobrushi n de - veloped thes e qualitie s an d attaine d a n extraordinar y dept h i n hi s "probabilistic " intuition. I n hi s las t year s a s a n undergraduate , R . L . Dobrushi n worke d unde r A. N. Kolmogorov o n certain problem s i n the theory o f Markov processe s an d wa s recommended t o graduate schoo l by the latter. I t shoul d be noted that i t wa s only as th e resul t o f grea t effort s an d obstinac y tha t A . N . Kolmogoro v succeede d t o have hi m accepte d i n th e graduat e program , becaus e i n that perio d (th e year s o f Stalin's governmental antisemiti c policies) ethnic Jews were ordinarily banned eve n from doin g undergraduate mathematics . Upon completio n o f hi s graduate wor k i n 1955 , R. L . Dobrushi n defende d hi s kandidaVs thesi s (PhD ) o n "Th e loca l limi t theore m fo r Marko v chains " an d ob - tained a positio n a t th e chai r o f probabilit y theor y o f the Mechanic s an d Math - ematics Departmen t o f Mosco w University , wher e h e taugh t unti l 1967 . I t wa s approximately a t thi s sam e tim e tha t R . L . Dobrushi n bega n workin g i n a ne w branch o f science : informatio n theory . Th e completio n o f hi s wor k i n tha t direc - tion wa s marke d b y hi s successfu l defens e i n 196 1 o f hi s doktorskaya dissertatio n (habilitation) o n "Informatio n theor y an d coding. " It shoul d b e noted tha t R . L . Dobrushin wa s always interested i n "physics " i n the broades t sens e o f thi s term , a s th e par t o f th e rea l worl d tha t coul d onl y b e described i n the language o f mathematics (say , the language o f probability theory ) and whic h woul d fill thi s languag e wit h ne w an d meaningfu l content . An d so , continuing t o wor k i n informatio n theory , R . L . Dobrushi n bega n hi s stud y o f statistical physics , which wa s to becom e hi s main field o f research unti l the en d o f his life . In 196 7 R. L . Dobrushi n wa s invite d t o hea d th e Laborator y o f Codin g The - ory a t th e Institut e o f Informatio n Transmissio n Problem s (IITP ) o f the Russia n Academy o f Sciences and lef t th e university, although he remained on e o f the head s of th e statistic s physic s semina r a t th e Mathematic s an d Mechanic s Departmen t there. R . L . Dobrushin's pedagogica l work als o continued a t the Physico-Technica l Institute, wher e h e lectured i n information theor y fo r man y years . xi xii A BRIE F BIOGRAPH Y

In his laboratory a t IITP R . L. Dobrushin brought together a remarkable grou p of mathematicians workin g actively in different area s o f this science. Here , becaus e of his taste fo r meaningfu l applications , he endeavored to interest the collaborator s in applied problems. I n this connection he always stressed that eac h specific applie d problem shoul d b e include d i n a mor e genera l scientifi c context , an d onl y the n could i t b e entirel y clarified . Afte r hi s death , th e laborator y tha t h e ha d heade d was renamed th e "Dobrushi n Mathematica l Laboratory. " Thi s nam e reflect s bot h Dobrushin's hug e mathematica l achievement s an d hi s outstandin g capabilitie s a s an organizer . R. L . Dobrushin' s mathematica l talen t wa s intimatel y connecte d t o hi s rar e pedagogical gift . Hi s lecture s an d semina r report s wer e alway s carefull y prepare d and significant , the y invariabl y gathere d a larg e numbe r o f listeners . H e ha d nu - merous pupils and collaborators, and i t was always instructive to talk mathematic s with him . In 199 5 R. L . Dobrushin wa s diagnosed wit h cancer , an d hi s health noticeabl y deteriorated fro m da y t o day . Bu t unti l th e ver y las t day s h e believe d i n hi s ow n recovery an d continue d t o wor k with the sam e intensity. H e died o n November 12 , 1995. Th e ur n wit h hi s remain s i s buried a t th e Vostryako v cemeter y i n Moscow . For his family, hi s friends, collaborators , and colleagues his death was a harsh blow , felt strongl y to this day . This book i s a tribute t o hi s memory .

Translated b y A . B . SOSSINSK Y Selected Title s i n Thi s Subserie s (Continued from, the front of this publication) Published Earlie r a s Advances i n Sovie t Mathematic s 21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4 20 Ft . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4 19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4 18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4 17 Dmitr y Fuchs , Editor , Unconventiona l Li e algebras, 199 3 16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M. Gelfan d seminar , Part s 1 and 2 , 1993 15 A . T . Fomenko , Editor , Minima l surfaces , 199 3 14 Yu . S . Il'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2 13 V . P . Maslo v an d S . N . Samborskii , Editors , Idempoten t analysis , 199 2 12 R . Z . Khasminskii , Editor , Topic s i n nonparametric estimation , 199 2 11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2 10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f global attractor s o f partia l differential equations , 199 2 9 A . M . Vershik , Editor , Representatio n theor y an d dynamica l systems , 199 2 8 E . B . Vinberg , Editor , Li e groups, their discret e subgroups , an d invarian t theory , 199 2 7 M . Sh . Birman , Editor , Estimate s an d asymptotic s fo r discret e spectr a o f integral an d differential equations , 199 1 6 A . T . Fomenko , Editor , Topologica l classificatio n o f integrable systems , 199 1 5 R . A . Minlos , Editor , Many-particl e Hamiltonians : spectr a an d scattering , 199 1 4 A . A . Suslin , Editor , Algebrai c if-theory , 199 1 3 Ya . G . Sinai , Editor , Dynamica l system s an d statistica l mechanics , 199 1 2 A . A . Kirillov , Editor , Topic s i n representation theory , 199 1 1 V . I . Arnold , Editor , Theor y o f singularities an d it s applications , 199 0