arXiv:cond-mat/0008310v1 [cond-mat.stat-mech] 22 Aug 2000 atce uvv ftebacigrt xed critical a exceeds rate branching value the the if Namely, survive particles. particles the of observed concentration be average can the thein transition varying phase the a When of probability consequence events. branching as branching site or a share jump to mentioned and try branch- they lattice a if other a with each particles of additional sites probability create ing neighboring can the one of each one on domly and Cardy by papers the see the T¨auber references in [3,4]). is (for studied This years extensively last been the etc. have BARWs strategies, the colonies, why biolog- do- reason their sites, represent or active can species atoms, walkers ical defects, the vortices, phenomena walls, ex- main these the In differ- systems of economical [2]. in models and biological behavior simplest chemical, physical, critical the ent exhibiting of processes one tinction as considered is n ean ai ntegah o 91] nother In magnetization [9–11]. spontaneous too the graphs of break- existence the symmetry the in continuous words, valid of remains absence ing the cri- recurrence for the that Mermin- terion un- shows the graphs general of to extension more theorem Wagner the a example, provides For graphs derstanding. on phenomena ical is struc- sites the the of in relevant. position longer graphs particles spatial no these the the of where for systems joints paths tureless The possible study by the joints. will characterized of define we graphs number paper regular uniform present random a the on Takayasu BARW In by the gasket [8]. Sierpinski Tretyakov sim- to and (MC) extended Carlo are Monte ulations The [3,4,6–8]. techniques ferent one-component the of [5]. most system in well processes as extinction class the belongs universality as (DP) state percolation absorbing directed the the to The to active time. the of from independent is transition which particles) (no state ing ngnrl h akr hneot atce)jm ran- jump particles) (henceforth walkers the general, In [1] (BARW) walk random annihilating branching The ti mhszdta h netgto fsm phys- some of investigation the that emphasized is It dif- by investigated well are BARWs the lattices the On P c httecnetaindcesslnal ihtebranchi the with linearly decreases concentration the that embcmszr if zero becomes term fnihos( neighbors of ASnmes 51.a 54.b 64.60.Ht 05.40.Fb, graph. 05.10.-a, same numbers: the PACS on walker single annihilation a mutual of features of returning probability the consideration into tews h ytmtnstwr h absorb- the toward tends system the otherwise , h rnhn niiaigrno aki tde narando a on studied is walk random annihilating branching The P rnhn niiaigrno ako admrglrgraph regular random on walk random annihilating Branching utemr,toprilsannihilate particles two Furthermore, . z .TeMneCrosmltosi gemn ihtegeneral the with agreement in simulations Carlo Monte The ). z eerhIsiuefrTcnclPyisadMtrasSci Materials and Physics Technical for Institute Research .Teefaue r ecie yamdfidma-edtheo mean-field modified a by described are features These 3. = ...4,H12 uaet Hungary Budapest, H-1525 49, P.O.B. coe 8 2018 28, October Gy¨orgy Szab´o 1 hscs h atcedsrbto sdsrbdb the by described is distribution In particle the [13]. straightforward case one-dimensional is this tree the technique locally of cluster a generalization dynamical For the structure methods. like different using by termined h vrg ocnrto fwles( walkers of concentration average the h ubro ie o large for sites average of with The number logarithmically them. the increases joining be- sites two path (number distance between shortest length distance A the the of as trees. steps) introduced of be to can similar sites two are tween graphs these cally uvv ntesnl cuidpisif pairs occupied single the on survive eut 486.Oraaye ilb ocnrtdo the on large concentrated sufficiently fea- be with the one-dimensional will graphs and the analyses random by Our loops described disjoint [4,8,6]. be results of can BARW set of a ture becomes graph the n h atce aihfor vanish particles the and niiaeec te evn nepyst behind. site empty an leaving particles is incomer other site and each neghboring resident annihilate chosen the randomly then the occupied already if cases, both In atceo n ftenihoigst ihaprobabil- a with site neighboring the of additional ity an one pro- creates evolution on elementary particle time particle chosen following The randomly the occu- A empty. be repeating cesses. or can by site particle governed each single is time a given by a pied neighbors) At (henceforth itself. sites excluding chosen randomly different, ossigof for consisting site starting the to walk. other returning annihila- random by of simple mutual affected a those their motion to not of the equals is probability if tion offspring the result, its then a and particles As parent graph. the walker single same of a the to the on mapped of be problem variation can The them meet. between offspring they distance its when and annihilated particle cru- a be a because will also BARW plays the walk to in random same role returning a the cial of of on walker recurrence probability random The single the graph. a to for point related starting is the graph a on grt for rate ng fteprn n t ffpigprilsuigthe using particles offspring its and parent the of h ttoaysaeo hssse scaatrzdby characterized is system this of state stationary The ntecs of case the In u netgto ilb ocnrtdo h graphs the on concentrated be will investigation Our P rjmst hsst wt rbblt 1 probability a (with site this to jumps or rp hs ie aeuiomnumber uniform have sites whose graph m z N ≥ hl h offiin ftelinear the of coefficient the while 4 z ie n ahst has site each and sites h rp ossso ijitpairs disjoint of consists graph the 1 = zdma-edaayi indicate analysis mean-field ized ence N > P ytkn explicitly taking ry [12]. ,wieteparticles the while 0, c N P htwl ede- be will that ) s z and .For 0. = onstoward joints z ≥ .Lo- 3. − z 2 = P ). configuration probabilities pk(n1,...,nk) (ni = 0 or 1) average concentration is illustrated in a log-log plot. As on the clusters of the neighboring k sites. Here we as- shown in Fig. 2 the MC data refer to a quadratic behav- sume that these quantities satisfy some symmetry (trans- ior for small P values, meanwhile the k-point approxima- lation, rotation, reflection) and compatibility relations. tions predict linear behavior with a coefficient decreasing The one-point configuration probabilities are directly re- when k is increased. lated to the average concentrations, namely, p1(1) = c 0.5 and p1(0) = 1 − c. Introducing an additional parameter q, the two-point configuration probabilities are given as p2(1, 1) = q, p2(1, 0) = p2(0, 1) = c − q and p2(0, 0) = 1 − 2c + q. Further parameters are required for k> 2. 0.4 In the present case the time variation of pk can be expressed by the terms of pk and pk+1. For example, 0.3

p˙1(1) = (1 − P )p1(1) − p2(1, 1)+ p2(0, 1) (1) c 0.2 where we have summed the contribution of all the el- ementary processes mentioned above. Notice that this equation is satisfied by the absorbing state (c = 0). 0.1 At the level of one-point approximation we assume that p2(n1,n2) = p1(n1)p1(n2). In this case the non-trivial stationary solution of Eq. (1) obeys a simple form, 0.0 0.0 0.5 1.0 P P c(1p) = (2) 2 FIG. 1. The average concentration of particles as a func- independent of z. At the level of k-point approximation tion of branching rate for z = 4. The symbols represent MC the corresponding set of equations is solved by using the data, the solid curves comes from the 1-, 2- and 5-point ap- proximations (from top to bottom) on clusters indicated at Bayesian relations (pk+1s are approximated by the prod- the top. uct of pk terms) [13]. In the two-point approximation the straightforward calculation gives the following stationary solution:

(2p) 2(z − 2) − (z − 3)P c = P 2 . (3) 4(z − 1) − 2zP +2P 10-1 This result refers to the absence of pair correlations in the limit z → ∞ as well as at P = 1 for any values of z. At higher levels the stationary solutions are evaluated 10-2 numerically. c In order to check these results we have performed MC simulations on random graphs with N = 500, 000 sites -3 varying the branching rate P for z = 4 and 3. The simu- 10 lations are started from a randomly half-filled graph and the concentration is obtained by averaging over 104 MC steps per particles after some thermalization. -4 10 -3 -2 -1 0 Figure 1 compares the MC data to the prediction of 10 10 10 10 k-point approximations for z = 4. Here the results of P 3- and 4-point approximations are omitted because their deviation from c(2p) is comparable to the line thickness. FIG. 2. Log-log plot of the average concentration of parti- In this case c = AP when P → 0. The A coefficients cles vs P if z = 3. The solid curves indicate the prediction obtained by MC simulation and 5-point approximation of generalized mean-field methods at the levels of 1-, 2-, 4-, and 6-point approximations (from top to bottom). The fitted are slightly different, namely, A(MC) = 0.250(2) and curve (dashed line) on the MC data (open diamonds) shows A(5p) =0.278(1). quadratic behavior. In the light of pair approximation [see Eq. (3)] better and better agreement is expected when increasing z. For Here it is worth mentioning that the MC data have z = 3, however, significant differences can be observed remained unchanged within the statistical error (compa- between the MC results and the prediction of k-point rable to the line thickness) when the random graph was approximations when P → 0. To magnify the discrep- generated in a different way. This investigation was mo- ancy between the two methods the P -dependence of the

2 tivated by the small word model suggested by Watts and (P = 0) when the particle concentration c(t) decreases Strogatz [14]. In this case the points with the first and monotonously. From Eqs. (4) and (5) we obtain that second joints form a single loop and the third ones are 1 chosen randomly. c(t)= (6) 2Q(z)t The trend in the prediction of k-point approximations implies the possibility that the coefficient of the linear when t → ∞. Figure 3 compares this result with the MC term goes to zero in the limit k → ∞. Now we introduce data as well as with the prediction of k-point approxi- a modified mean-field theory taking explicitly into con- mations obtained by numerical integration of the corre- sideration the mutual annihilation of the parent and his sponding set of differential equations for z = 3 and 4. offspring as mentioned above. It will be shown that the In the MC simulations the system is started from a half- vanishment of the linear term is directly related to fact filled random initial state for N = 106 and the data are that half of the branching process yields mutual annihi- averaged over 50 runs. The classical mean-field (1-point) lation. approximation corresponds to the choice of Q(z) = 1. In this description the R(z) probability of this mutual From the k-point approximations, however, we can de- annihilation process is approximated with the probability duce smaller Q(z) values. For example, in the pair ap- of returning to the starting site for a single random walker proximation Q(2p)(4) = 3/4 and Q(2p)(3) = 2/3. These on the same graph. This simplification is exact in the estimations can be improved if we choose larger and limits P and c → 0 when the random walks of the parent larger clusters as illustrated in Fig. 3. We have obtained and his offspring are practically not affected by other Q(5p)(4) = 0.692(1) for z = 4 and Q(6p)(3) = 0.534(1) for particles. The appearance of additional offsprings during z = 3 using 5- and 6-point approximations respectively. the recurrence is considered a second order process whose At the same time, the numerical fitting to the MC data effect will be discussed below. results in Q(MC)(4) = 0.67(2) and Q(MC)(3) = 0.50(2). The main advance of this approach is that we can use Notice that the predictions of k-point approximations the exact results obtained for the simple random walk on tend slowly toward the MC values. According to the nu- the Bethe lattices which are locally similar to the present merical integration in k-point approximations the asymp- random graphs. Hughes and Sahimi [15] and Cassi [16] totic behavior is independent of the initial concentration have obtained that R(z)=1/(z − 1) and the average c(0). number of steps to return to the starting site is τ(z) = 2(z − 1)/(z − 2). The value of τ(z) indicates that the particle visits only a few sites before its recurrence and 1.0 during this short period it is not capable to distinguish the Bethe lattices from the present random graphs due to the absence (low probability) of loops. In fact, this is the reason why our analysis is restricted to large N. As a result of the mutual annihilation, R(z) portion of the branching process can be considered as spontaneous tc(t) 0.5 annihilations. On the other hand, 1 − R(z) portion of the branching events results in “independent” particles. These features can be easily built into a modified mean- field theory which obeys the following form:

p˙1(1) = − (1 − P )p1(1) − p2(1, 1) − P R(z)p2(1, 0) 0.0 0 1 2 3 4 + [1 − P R(z)]p2(0, 1) . (4) 10 10 10 10 10 t [MCS] The two-point configuration probabilities are also af- fected by the random walk itself, because a given particle FIG. 3. Time-dependence of average concentration multi- leaves an empty site behind when it steps to one of the plied by time for P = 0 and c(0) = 1/2. The open squares (diamonds) represent MC data for z = 3(z = 4). The neighboring sites. Consequently, the value of p2(1, 0) is larger than those predicted by the mean-field theory. The upper (lower) solid curve indicates the prediction of 6-point above mentioned techniques confirm that this correlation (5-point) approximation for z =3(z = 4). The dashed line can be well described by a simple parameter defined as shows the prediction of classical mean-field approximation. The fitted asymptotic functions are represented by dotted lines. p2(1, 1) = Q(z)p1(1)p1(1) (5) where Q(z) ≤ 1 and the remaining two-point configu- Using the expression (5) the nontrivial stationary so- ration probabilities are determined by the compatibility lution of Eq. (4) obeys the following simple form: conditions. The value of Q(z) is related to the asymp- 1 − 2R(z) c = P . (7) totic time-dependence for the annihilating random walk 2[1 − P R(z)]Q(z)

3 Evidently, beside this expression Eq. (7) has a trivial [1] M. Bramson and L. Gray, Z. Wahrsch. verw. Gebiete 68, solution (c = 0) which remains the only one for R> 1/2. 447 (1985). In the limit P → 0 the leading term of Eq. (7) can be [2] J. Marro and R. Dickman, Nonequilibrium Phase Tran- expressed as c = P [1 − 2R(z)]/2Q(z). This prediction sitions in Lattice Models (Cambridge University Press, coincides with the above mentioned MC result for z =4 Cambridge, 1998). if Q(MC)(4) is substituted for Q(z). [3] J. L. Cardy and U. C. T¨auber, Phys. Rev. Lett. 77, 4780 According to Eq. (7) the average concentration be- (1996). 90 comes zero for z = 3. This is a consequence of the fact [4] J. L. Cardy and U. C. T¨auber, J. Stat. Phys. , 1 (1998). 42 that the rare branching processes do not modify the av- [5] H. K. Janssen, Z. Phys. B , 151 (1981); P. Grassberger, Z. Phys. 47, 365 (1982). erage number of walkers because these events result in 26 50 zero or two walkers with the same probability on a longer [6] I. Jensen, J. Phys. A , 3921 (1993); Phys. Rev. E , 3623 (1994). time scale. There exists, however, a second order term [7] N. Inui and A. Yu. Tretyakov, Phys. Rev. Lett. 80, 5148 neglected above which is responsible for the quadratic be- (1998). havior. Namely, during the recurrence one of the (parent [8] H. Takayasu and A. Yu. Tretyakov, Phys. Rev. Lett. 68, and offspring) particles can create an additional walker 2 3060 (1992). with a probability proportional to P R(z)[τ(z) − 1] and [9] D. Cassi, Phys. Rev. Lett. 68, 3631 (1992). this event reduces the rate of spontaneous annihilation [10] F. Merkl and H. Wagner, J. Stat. Phys. 75, 153 (1994). [P R(z)] in Eqs. (4) and (7). Unfortunately, this rough [11] R. Burioni, D. Cassi, and A. Vezzani, Phys. Rev. E 60, approach can not reproduce accurately the coefficient of 1500 (1999). the quadratic term because of the simplification we used [12] B. Bollob´as, Random Graphs (Academic Press, New in the derivation of Eq. (4). York, 1985). The present analysis indicates that the probability of [13] H. A. Gutowitz, J. D. Victor, and B. W. Knight, Physica recurrence of a single walker, the long time behavior of 28D, 18 (1987). the time-dependent concentration for the annihilating [14] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). random walks and the stationary concentration for the [15] B. D. Hughes and M. Sahimi, J. Stat. Phys. 29, 781 BARW in the limit P → 0 are strongly related to each (1982). other. The modified mean-field theory suggests a way to [16] D. Cassi, Europhys. Lett. 9, 627 (1989). describe this relation on random regular graphs for suffi- [17] Some quantitative features of random walk on Bethe and ciently large number of sites. The relation is based on two cubic lattices (R and τ) are compared in Ref. [15]. 5 simple conditions. Namely, R(z) ≤ 1/2 and τ(z) is finite. [18] P. Erd˝os, A. R´enyi, Publ. Math. Inst. Hun. Acad. Sci. , These conditions are also fulfilled on the cubic lattices for 17 (1960). 286 dimensions d> 4 [17] where the rigorous analysis predicts [19] A.-L. Barab´asi and R. Albert, Science , 509 (1999). mean-field type behavior [4]. Conversely, the modified mean-field theory is not adequate for lower dimensions (d ≤ 3) because τ = ∞. It is expected, however, that the present analysis can be extended to the investigation of BARW on such random graphs characterized with differ- ent probability distributions of connectivity [18,19]. The present random graphs for z = 3 represent a particular situation where the coefficient of the linear term vanishes as a consequence of R(3) = 1/2. The solutions of the modified mean-field theory imply the the possibility of another interesting situation for those above mentioned random graphs [12,14,18,19] where the average value of R is larger than 1/2. In this case the particles can sur- vive if the branching rate exceeds a treshold value as it happens on the one- and two-dimensional lattices.

ACKNOWLEDGMENTS

Support from the Hungarian National Research Fund (T-23552) is acknowledged.

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