Multi-particle processes with reinforcements.

Transcription

1 Muli-paricle processes wih reinforcemens. Yevgeniy Kovchegov Deparmen of Mahemaics, Oregon Sae Universiy Corvallis, OR , USA Absrac We consider a muli-paricle generalizaion of linear edge-reinforced random walk (ERRW). We observe ha in absence of exchangeabiliy, new echniques are needed in order o sudy he muli-paricle model. We describe an unusual coupling consrucion associaed wih he wo-poin edge-reinforced process on Z and prove a form of recurrence: he wo paricles mee infiniely ofen a.s. 1 1 Inroducion The edge-reinforced random walk was firs inroduced in [2] and [5]. In his paper we will sudy linear muli-paricle edge-reinforced processes on Z. In he original edge-reinforced random walk model, each edge of a locally finie non-direced graph is iniially assigned weigh a > 0. Wih each sep, he paricle jumps o a neares-neighbor verex. The probabiliy of he jump equals o he fracion of he weigh aached o he raversed edge in he oal sum of he weighs of he edges coming ou of he verex where he paricle is locaed prior o he jump. Each ime an edge is raversed, is weigh is increased by 1. In oher words, he linear edge-reinforced random walk is a random walk on a weighed graph, where he weigh of an edge is increased by one each ime i is being raversed. One of he mos imporan open problems in he heory of reinforced random walks is ha of checking if he linear edge-reinforced random walk is recurren on Z d for dimensions d 2. Linear edge-reinforced random walk is exchangeable making he model an imporan example for applying he heorem of de Finei (see [7]) and is generalizaions (see [4], [5]). The hisory and some of he mos imporan resuls in reinforced processes can be found in [5], [3], [10], [8], [1], [12], [11] and references herein. This research was suppored in par by NSF VIGRE Gran DMS a UCLA 1 Keywords: edge-reinforced processes, random walk, random environmen, urn model, coupling mehod, recurrence AMS Subjec Classificaion: 60C05, 60G07, 60G09, 60K37 1

2 Muli-paricle processes wih reinforcemens 2 In his paper we will consider a muli-paricle modificaion of he edge-reinforced random walk model similar o some of he reinforced processes sudied in [11]. We le he walker (or paricle) in he edge-reinforced random walk model wai an independen exponenial ime wih rae one beween he jumps. So, he walker jumps from a sie o a near by sie wih raes equal o he corresponding raios. Now we are ready o define an n-poin process η = {η 1 (),..., η n ()}, where all n paricles ravel along he edges of a graph G, jumping from a sie o a neighboring sie in S, he se of all sies. Now, le W (e 1 ),..., W (e k ) be he weighs assigned o all k edges e 1,..., e k coming ou of a given sie v S a ime. Once again, he iniial weighs are all assigned o be equal o a > 0, i.e. W 0 (e 1 ) = W 0 (e 2 ) = = W 0 (e k ) = a. If one of he paricles, say η j, is a sie v a when is exponenial clock rings, hen he paricle W (e i ) W (e 1 )+ +W (e k ) raverses e i (1 i k) wih he rae =. In which case he corresponding edge weigh increases by 1, i.e. W (e i ) = W (e i ) + 1. The recurrence/ransience quesions arising in his more general model are as imporan as he corresponding quesions in he heory of one-aricle edge-reinforced random walks. The edge-reinforced process on Z wih drif > 0 can be defined in he following way: if a paricle is a sie v Z a he jump ime, hen he probabiliy of he paricle jumping o v + 1 is W (v, v + 1) + W (v 1, v) + W (v, v + 1) + while he probabiliy of i jumping o v 1 is W (v 1, v) W (v 1, v) + W (v, v + 1) +. On rees, he edge-reinforced process wih a oward-he-roo drif > 0 can be saed accordingly. We will concenrae on he mos basic case of muli-paricle reinforced processes: he wo poin reinforced process η = {η 1 (), η 2 ()} on Z wih drif 0. We will describe an unusual coupling consrucion associaed wih he process and as a consequence prove he recurrence of (η 2 () η 1 ()) whenever 0 < 1. In he case of a wo-paricle process on Z, one of he wo paricles is locaed o he lef of anoher, excep for he imes when boh paricles are a he same sie. We will denoe by l he locaion of he lef paricle, and by r he locaion of he righ paricle a ime. When r = l here is no need o disinguish beween he lef and he righ paricles. The difference becomes apparen only when one of he paricles leaves he sie. So, η = {l, r }, and here is he main resul of his paper: Theorem 1. For all 0 < 1 and a > 0, (r l ) is recurren. Le us begin by reviewing he Polya s urn model. The urn iniially conains R 0 marbles of red color, and B 0 marbles of blue color. We fix a posiive ineger number D. A marble is randomly and uniformly drawn from he urn, reurned, and D marbles of he same color are added. Le R n and B n be respecively he number of red and blue marbles in he urn afer n drawings, and le ρ n = Rn R n+b n be he fracion of he red marbles in he urn afer n drawings. I is easy o show ha ρ n is a maringale, and herefore, by maringale convergence heorem,

3 Muli-paricle processes wih reinforcemens 3 converges o a random variable. Tha random variable ρ is in urn shown o be a bea random variable wih parameers R 0 and B 0, i.e. one wih bea densiy funcion D D 1 )xr0 β( R 0, B D 1 (1 x) B 0 D 1, (1) 0 D D where β(a, b) = Γ(a+b). One can check ha he urn model is exchangeable (see [7]), ha is Γ(a)Γ(b) if one permues he resuls of m consecuive drawings, he probabiliy of he oucome does no change. By de Finei s heorem, condiioned on ρ, he resuls of he drawings are independen Bernoulli rials, where each ime a red marble is seleced wih probabiliy ρ and a blue marble is seleced wih probabiliy 1 ρ. The model rivially exends o he case when R 0, B 0 and D are posiive real numbers, as well as when here are more han wo differen ypes of marbles. For insance, consider he case when here are hree ypes of marbles, red, blue and green, in he urn. If we sar wih he amouns R 0, B 0 and G 0 of respecively red, blue and green marbles, hen he limiing fracions vecor will be a Dirichle disribued random vecor, i.e. he cumulaive densiy funcion for he limiing fracions of red and blue marbles will be Γ ( R 0 +B 0 +G 0 ) D f(x, y) = Γ ( ) ( R 0 D Γ B0 ) ( D Γ G0 )x R 0 D 1 y B 0 D 1 (1 x y) G 0 D 1 if x > 0, y > 0 and x + y < 1, D where he above densiy is derived by applying (1) wice. See [7] for basic facs on exchangeabiliy, he Polya s urn model and a simple version of de Finei s heorem. A simple proof of he convergence o bea disribuion can be found in [15]. Polya s urns were used o sudy linear edge-reinforced random walks (see [12]). There, if he walk lives on an acyclic graph, say Z, we can assign a Polya s urn for each sie. When he walker is a sie v, we do a drawing from he urn associaed wih v, where he number of he red (respecively blue) marbles in he urn is equal o he weigh aached o he edge [v 1, v] (respecively [v, v + 1]) a he ime. If a red marble is drawn, he walker jumps o v 1 and we add D = 2 red marbles o he urn associaed wih v. Similarly, if a blue marble is drawn, he walker jumps o v + 1 and we add D = 2 blue marbles. We do so because he graph is acyclic: if he paricle ever reurns o he verex, i will be from he same direcion i ook when i lef he verex. For example, if he red marble is drawn, he walker will raverse he edge [v 1, v] wice before reurning o v, hus increasing he weigh of he edge exacly by D = 2. The iniial condiions [R 0 (v), B 0 (v)] a an urn associaed wih sie v should be se equal o he weighs aached o he edges [v 1, v] and [v, v + 1] respecively a he ime of he firs arrival o v. As an example, consider he edge-reinforced random walk on Z ha begins a sie 0. There he correc iniial condiions for a Polya s urn assigned o sie v Z should be se equal o [a, a + 1] if v < 0, [R 0 (v), B 0 (v)] = [a, a] if v = 0, [a + 1, a] if v > 0.

4 Muli-paricle processes wih reinforcemens 4 Wha follows is ha one can do an infinie number of drawings independenly for each of he Polya s urns associaed wih he verices of an acyclic graph before he walk begins, hus compleely predeermining he rajecory of he walker. Now, he exchangeabiliy propery of Polya s urns and de Finei s heorem menioned above allows one o resae he edge-reinforced random walk as a random walk in random environmen (RWRE), where he environmen is disribued as he limiing bea random variables obained for Polya s urn processes associaed wih each verex of he acyclic graph. Afer ha, oher echniques such as large deviaions are of use in answering he corresponding recurrence/ransience quesions for he RWRE model (see [12]). Does he same approach work for he wo poin process η = {l, r } on Z? The answer is no. Consider he case when he drif = 0. Suppose here is an urn a each verex of Z. Suppose a verex v is visied by he righ paricle r, and he drawing was done from he urn associaed wih v and a blue marble was seleced, so ha he righ paricle r jumps o v + 1. We canno add D = 2 blue marbles ino he urn, as i could happen ha he lef paricle l arrives o he urn from (, v 1] before he righ paricle r reurns o v from [v + 1, ). In he laer case, here will be more blue marbles han he weigh amoun aached o he edge [v, v + 1] on he righ and he raes will no agree. In oher words, he represenaion wih Polya s urns and similar approaches will no work because he wo-poin linear edge-reinforced process is nonexchangeable. The non-exchangeabiliy of he process was he main obsacle for sudying i as well as for proving Theorem 1. 2 The Polya s urn modified Alhough he represenaion wih classical Polya s urns fails for he wo-poin process η = {l, r }, here is a way o modify i. Suppose ha a each verex of Z, he associaed urn conains no only he red and blue marbles, bu also a special marble, called magic marble, such ha when he lef paricle l arrives o he sie, he magic marble becomes red, while when he righ paricle r arrives o he sie, he magic marble becomes blue. Each urn will conain exacly one magic marble in addiion o red and blue marbles. When magic marble is seleced, wo marbles of he color assumed by he magic marble will be added ino he urn. Once again he paricles move according o he colors of marbles seleced from he urns. In oher words, if he magic marble is seleced when i is red, wo more red marbles will be added o he urn and he paricle will jump lef. Similarly, if he magic marble is seleced when i is blue, wo more blue marbles will be added o he urn and he paricle will jump righ. Le R (v) and B (v) denoe respecively he number of red and he number of blue marbles inside an urn associaed wih sie v Z, a ime. The iniial number of red and blue marbles, R 0 (v) and B 0 (v), ogeher wih he magic marble mus represen he corresponding weighs assigned o edges [v 1, v] and [v, v + 1] a he ime of he firs arrival o v by any of he wo paricles. The lef paricle is he firs o visi he sies o he lef of l 0, i.e. all v < l 0, and he righ paricle is firs o visi he sies o he righ of r 0. Hence, for all a > 0, he following mus be he iniial configuraion of red and blue marbles assigned o

5 Muli-paricle processes wih reinforcemens 5 he urns associaed wih sies in Z: [a 1, 1 + a + ] if v < l 0, [a 1, a + ] if v = l 0, [R 0 (v), B 0 (v)] = [a, a + ] if l 0 < v < r 0, [a, a 1 + ] if v = r 0, [a + 1, a 1 + ] if r 0 < v (2) plus a magic marble in every urn. Here we can allow a 1 < 0 since here is also a magic marble in he urn, which is red when he lef paricle is a he sie, and blue when he righ paricle is a he sie. We now explain he reason why he magic marble was inroduced. Firs we check ha he above urn represenaion produces correc raes up unil he firs recurrence ime τ 1 := min{ : l = r }. We consider he case when he righ paricle depars from sie v a jump ime < τ 1. Suppose ha he nex arrival o v happens before τ 1, hen here are hree possible scenarios. Case I: he righ paricle jumps o he lef, and reurns o v before he lef paricle arrives. So r = v, r = v 1, l < v, and we need o add wo red marbles ino he urn, i.e. R (v) = R (v) + 2. The magic marble says blue, and as i was he case wih one paricle ERRW model, he raes agree. Case II: he righ paricle jumps o he righ, bu reurns o v before he lef paricle arrives. Tha is r = v, r = v + 1, l < v and we need o add wo blue marbles ino he urn, i.e. B (v) = B (v) + 2. The raes agree since he righ paricle reurns o v before he nex visi o v by he lef paricle. Again, he magic marble says blue, and as i was he case wih one paricle ERRW model, he raes agree. Case III: he righ paricle jumps o he righ, and he lef paricle arrives o v before he righ paricle reurns from [v + 1, + ). This is he case where he chameleon propery of he magic marble is used. Once again r = v, r = v + 1, l < v and we need o add wo blue marbles ino he urn, i.e. B (v) = B (v) + 2. Before he deparure of he righ paricle from sie v a ime, and he weigh assigned o [v 1, v] was = R (v) he weigh assigned o [v, v + 1] was = B (v) + 1 as he magic marble was blue in he presence of he righ paricle. When he lef paricle jumps o v from v 1 a ime 1 (, τ 1 ) before he reurn of he righ paricle, he magic marble re-colors ino red, and and he weigh assigned o [v 1, v] = R 1 (v) + 1 = R (v) + 1 he weigh assigned o [v, v + 1] = B 1 (v) = B (v) = B (v) + 2. One can see ha he weighs are correc since each edge [v 1, v] and [v, v + 1] was raversed exacly once.

6 Muli-paricle processes wih reinforcemens 6 Figure 1: Above: = 0, he righ paricle is a sie v, W (v 1, v) = 3 and W (v, v + 1) = 2. The urn associaed wih v conains hree red, one blue and one magic marble. The magic marble is emporarily colored in blue as he righ paricle is presen. Below: wih probabiliy 2, eiher blue or magic marble was seleced, he righ paricle jumps o v + 1 and wo blue 5 marbles are added ino he urn. The lef paricle arrives o v from he lef while he righ paricle is sill a v + 1. The weigh ha corresponds o [v 1, v] is equal o 4, while he weigh of [v, v + 1] is equal o 3. There are hree red, one magic and hree blue marbles in he urn. However, he magic marble is emporarily red as he lef paricle is presen a sie v.

7 Muli-paricle processes wih reinforcemens 7 Tha explains why adding magic marble works. The case when he lef paricle is a sie v can be checked by he analogy wih he case above. See Figure 1 for a visual example. Observe ha he above coupling of he urn process wih η works only up unil ime τ 1 = min{ : l = r }, he firs ime ha he paricles mee. Now, we need o show ha he paricles mee a leas once. Therefore before proving Theorem 1, we will need o prove he following one-ime recurrence resul: Theorem 2. For all 0 < 1 and all a > 0, τ 1 <. Since he above urn process is coupled wih he above described urn process unil he decoupling ime τ 1, i suffices o prove Theorem 2 for he urn process. Laer i will be shown ha he consrucion will also imply he full recurrence, i.e. Theorem 1. 3 Recurrence via coupling. The urn consrucion consrucion defined in he preceding secion deermines η = (l, r ) for 0 τ 1. Here we le q l (v) and p l (v) be respecively he lef and he righ jump raes for he lef paricle a sie v. We also denoe by q r (v) and p r (v) respecively he lef and he righ jump raes for he righ paricle a sie v. By consrucion, q l (v) = R (v) + 1 R (v) + B (v) + 1 and p l (v) = B (v) R (v) + B (v) + 1, and similarly, q r (v) = R (v) R (v) + B (v) + 1 and p r (v) = B (v) + 1 R (v) + B (v) + 1. We recall ha in he case of edge-reinforced random walks on Z, he drawings predeermined he oucome of he whole process. There he resuls of all drawings from he urns associaed wih all he verices of he graph deermined uniquely he rajecory of he walker. In he one-paricle case one deermines he limiing fracions of blue marbles for all sies in he form of respecive independen bea random variables. Then one inerpres he walk as a birh and deah chain wih hese raes. We wan o implemen a similar rick for he wo-poin edge-reinforced process. We will embed he urn process defined in he preceding secion ino a hree-color Polya s urn process. 3.1 Magic family For each sie v and he urn associaed wih v, we define he magic family as all he marbles ha were added as he resul of selecing he magic marble, plus he magic marble iself. Here is how he magic family is consruced: a he beginning he magic family consiss of only he magic marble iself. When he magic marble is seleced from he urn for he firs ime, and wo marbles (of one of he wo colors) are added o he urn, we include he wo ino he magic family. Each ime a marble from he magic family is seleced and wo new marbles are added, we le he wo marbles ino he magic family regardless of heir color.

8 Muli-paricle processes wih reinforcemens 8 All he red marbles ha are no in he magic family will be called pure red, and all he blue marbles ha are no in he magic family will be called pure blue. Observe ha for each sie v, he urn associaed wih v is a Polya s urn wih respec o hree ypes of marbles: pure red, pure blue and he marbles in he magic family. Each ime when a pure red marble is seleced, wo more pure red marbles are added o he urn. Same is rue for pure blue marbles. See Figure 2 for a visual example. Figure 2: Suppose a = 2, = 0 and v > r 0. We begin wih hree red, one blue and one magic marble inside he urn associaed wih sie v. The righ paricle was a v a he ime of he firs drawing. Afer waiing for he jump ime wih rae one, a pure blue marble is seleced, wo more pure blue marbles are added o he urn and he righ paricle jumps o v +1. Nex, he lef paricle arrives from v 1, and he magic marble assumes red color. The magic marble is seleced in he second drawing, wo magic family red marbles are added o he urn and he lef paricle jumps back o v 1. The righ paricle arrives from he righ for he hird drawing, he magic marble is seleced wih probabiliy 1, in which case wo magic 9 family blue marbles are added and he righ paricle jumps o sie v + 1. The lef paricle arrives from v 1, a magic family red marble is seleced, wo more magic family red marbles are added, and he lef paricle jumps back o v 1. For each sie v, we le R n (v), Bn (v) and M n (v) denoe respecively he number of pure red marbles, he number of pure blue marbles and he number of marbles in he magic family inside he urn associaed wih sie v afer n drawings. The proporion vecor of pure red

9 Muli-paricle processes wih reinforcemens 9 marbles, marbles in he magic family and pure blue marbles [ Rn (v), M n (v), B n (v)] /( R n (v) + M n (v) + B n (v)) converges o a Dirichle random vecor wih parameers R 0(v), 1 and B 0(v) We observe ha afer n drawings, Mn (v) 1 marbles in he magic family are of eiher red or blue color. Therefore R n (v) R n (v) and B n (v) B n (v). Le B(α, β) denoe he bea disribuion wih parameers α > 0 and β > 0. If for each v we define p l P olya (v) as he limiing fracion of pure blue marbles, hen p l P olya (v) will be a bea random variable wih parameers B 0(v) and R 0(v)+1 (. Looking back a (2), one can wrie down he corresponding 2 ) 2 B B0 (v), R 0(v)+1 disribuion of p l 2 2 P olya (v) for each v Z. For a 1, and for a > 1, p l P olya(v) is p l P olya(v) is B( a+1+, a 2 ) if v < l 2 0, B( a+, a) if v = l 2 2 0, B( a+, a+1) if l < v < r 0, 0 if r 0 v B( a+1+, a) if v < l 2 2 0, B( a+, a) if v = l 2 2 0, B( a+, a+1) if l < v < r 0, B( a 1+, a ) if v = r 0, B( a 1+, 1 + a) if r < v. Similarly, qp r olya (v) defined as he limiing fracion of pure red marbles in he urn will be a bea random variable wih parameers R 0(v) and B 0(v)+1. So for a 1, 2 2 and for a > 1, q r P olya(v) is q r P olya(v) is 0 if v l 0, B( a, a+1+ ) if l < v < r 0, B( a, a+ ) if v = r 2 2 0, B( a+1, a+ ) if r < v. B( a 1, 1 + a+ ) if v < l 2 2 0, B( a 1, a+1+ ) if v = l 2 2 0, B( a, a+1+ ) if l < v < r 0, B( a 2, a+ ) if v = r 2 0, B( a+1, a+ ) if r < v. Le q l P olya := 1 pl P olya and pr P olya := 1 qr P olya. Then he pairs (ql P olya (v), pl P olya (v)) v Z and (q r P olya (v), pr P olya (v)) v Z can be viewed as wo dependen random environmens. We define

10 Muli-paricle processes wih reinforcemens 10 as a random walk in he random environmen (qp l olya (v), pl P olya (v)) v Z ha sars a l P olya 0 = l 0 and jumps from v o v + 1 wih rae p l P olya (v) or o v 1 wih rae ql P olya (v). Similarly, we define r P olya as a random walk in he random environmen (qp r olya (v), pr P olya (v)) v Z ha sars a r P olya 0 = r 0 and jumps from v o v + 1 wih rae p r P olya (v) or o v 1 wih rae qp r olya (v). We observe ha condiioned on he wo dependen environmens, he wo random walks, l P olya and r P olya, can coexis as wo independen birh and deah chains. In he nex subsecion we will couple {l P olya, r P olya } wih {l, r } so ha l P olya l P olya l r r P olya for 0 τ 1. Then showing he recurrence of (r P olya l P olya ) will prove Theorem 2. Here is he heurisic explanaion for he coupling consrucion o follow. If in he original urn model {l, r }, we subsiue he magic marble wih a red marble in every urn, hen he lef paricle process l will be disribued as he random walk in random environmen l P olya. If in urn we subsiue he magic marble wih a blue marble in every urn, hen he righ paricle process will have he same disribuion as r P olya. Observe ha his heurisics can be generalized o work in he case of more han wo paricles, e.g. hree-paricle linear edge-reinforced processes on Z. 3.2 Coupling wih RWRE We noice ha he process {l, r } can be predeermined by he resuls of all drawings from he Polya s urns wih hree ypes of marbles: pure red, pure blue and magic family. We can firs do he drawings from he urns associaed wih all he sies in Z, deermining [ R n (v), M n (v), B n (v)] n=0,1,2,... for all v and he limiing fracions {qp r olya (v), pl P olya (v)} v Z of pure red and pure blue marbles. By de Finei s heorem, condiioned on qp r olya (v), pl P olya (v), [ R n (v), M n (v), B n (v)] n=0,1,2,... are deermined as independen rials wih probabiliies qp r olya (v) and p r P olya (v) for pure red and pure blue marbles respecively. When one of he wo paricles visis sie v, he resuls [ R n (v), M n (v), B n (v)] of he n-h drawing (if i is he ime of n-h deparure from he sie) deermine he desinaion sie and he coloring of he marbles in he magic family. We recall ha in he Polya s urn model he limiing fracion of marbles of one color is a paricular bea random variable and he densiy funcion f {q r P olya (v),p l P olya (v)} (x, y) for he pair of limiing fracions qp r olya (v) and pl P olya (v) is Dirichle ) f {q r P olya (v),p l P olya (v)} (x, y) = if x > 0, y > 0 and x + y 1. Γ Γ ( R0 (v) 2 ( R0 (v)+b 0 (v)+1 ) Γ 2 ( B0 (v) 2 ) Γ ( )x R0(v) 2 1 y B 0 (v) 2 1 (1 x y) Now we consruc he coupled process {l P olya, l, r, r P olya }: We condiion on he Dirichle variables {q r P olya (v), pl P olya (v)} v Z.

11 Muli-paricle processes wih reinforcemens 11 We begin wih l P olya 0 = l 0 < r 0 = r P olya 0. The rajecories of l and r are deermined by drawings from he urns. l P olya and r P olya are independen birh and deah chains wih corresponding probabiliies (qp l olya (v), pl P olya (v)) v Z and (qp r olya (v), pr P olya (v)) v Z ha move independenly of each oher and of l and r excep for he imes when l P olya = l or r P olya = r. When he lef paricles l and l P olya happen o be a he same sie v, hey wai for he deparure ime wih rae one. Then he resuls of he nex drawing from he corresponding urn are sudied. If he seleced marble is red or magic marble, boh paricles jump o v 1. However, if he seleced marble is blue, he lef paricle l jumps o v + 1, while l P olya jumps o v + 1 only if he marble is pure blue, and jumps o v 1 oherwise. The probabiliy ha l P olya jumps o v + 1 is equal o p l P olya (v), and each such drawing is independen of all he oher drawings. l P olya will sill be a random walk in random environmen (q l P olya (v), pl P olya (v)) v Z, while l P olya l is preserved. Similarly, when he righ paricles r and r P olya happen o be a he same sie v, hey wai for he deparure ime wih rae one. Then he resuls of he nex drawing are sudied. If he seleced marble is a blue or a magic marble, boh paricles jump o v+1. If he marble is red, he righ paricle r jumps o v 1, while r P olya jumps o v 1 only if he marble is pure red, and jumps o v + 1 oherwise. The probabiliy ha r P olya jumps o v 1 is equal o qp r olya (v), and each drawing is independen of he ohers. r P olya r r P olya will sill be a random walk in random environmen (qp r olya (v), pr P olya (v)) v Z, while is preserved. In he above coupled process, condiioned on he environmens, he independence of l P olya is preserved and and r P olya l P olya Thus showing he recurrence of (r P olya l r r P olya. l P olya ) is enough o prove Theorem 2. Some heory of RWREs: a RWRE on Z wih he righ jump probabiliy p(v) chosen o be B( a+1, a+ ) disribued a all sies v is ransien o he righ whenever 0 < 1, explaining 2 2 he bound on he drif in) Theorem 1. In general, RWRE on Z is a.s. ransien o he righ if and only if E[log ] > 0 (see [14] for he proof, and [13], [9] and references herein ( p(v) 1 p(v) for more on he subjec). No surprisingly he RWRE wih he environmen {p(v)} v Z

12 Muli-paricle processes wih reinforcemens 12 independenly B(α 1, α 2 ) disribued wih α 1 > α 2 is a.s. ransien o he righ: ( p(v) ) E[log ] = 1 p(v) where we subsiue α 1 > 1, and E [ 1 p(v) p(v) = 1 β(α 1, α 2 ) 4 β(α 1, α 2 ) 1 0 = 23 α 1 α 2 β(α 1, α 2 ) > 0, 0 ( x log 1 x ) x α 1 1 (1 x) α 2 1 dx se 2α 1s (1 + e 2s ) (α 1+α 2 ) ds x = 1 x e2s. In he above general case, E ] = + if α 1 1. s sinh((α 1 α 2 )s)(cosh s) (α 1+α 2 ) ds [ ] 1 p(v) = β(α 1 1,α 2 +1) p(v) β(α 1,α 2 = α 2 if ) α 1 1 [ ] The expeced ime of reurn is a.s. finie only when E 1 p(v) < 1 (see [14]). Tha is p(v) only when α 1 > 1 + α 2. Now, if one considers a RWRE on Z wih independen righ jump probabiliies {p(v)} v Z, each B( a+1, a+ ) disribued, he recurrence ime is infinie since 2 2 a+1 < 1 + a+ for > The above implies ha he expeced ime of reurn o sie v for r P olya is infinie if v < r P olya and is finie if r P olya < v. Similarly, he expeced ime of reurn o sie v for l P olya is infinie if l P olya < v and is finie if v < l P olya. 3.3 Proof of Theorem 2. In his subsecion we will prove Theorem 2. Proof of Theorem 2: Recall ha he environmens {(p l P olya (v), ql P olya (v)} v Z and {(p r P olya (v), qr P olya (v)} v Z of l P olya and r P olya are dependen, bu condiioned on he environmens, he walks l P olya and r P olya are independenly. We claim ha, even hough he expeced ime of reurn of (r P olya l P olya ) o zero is infinie, (r P olya l P olya ) is recurren. The problem can be summarized by he following more general lemma. Lemma 1. Le p 1 (1), p 1 (2),... be i.i.d. random variables defined on (0, 1) wih [ ( )] 1 p1 (i) µ 1 = E log > 0, p 1 (i) e.g. p 1 (i) B(a 1, b 1 ) for 0 < a 1 < b 1, and le p 2 (1), p 2 (2),... be i.i.d. random variables defined on (0, 1) wih [ ( )] 1 p2 (i) µ 2 = E log > 0, p 2 (i) e.g. p 2 (i) B(a 2, b 2 ) for 0 < a 2 < b 2. Also le p 1 (0) = p 2 (0) = 1. If p 1 (0), p 1 (1), p 1 (2),... are he forward raes for he birh-and-deah chain Z r, and p 2 (0), p 2 (1), p 2 (2),... are he forward raes for he birh-and-deah chain Z r, hen he wo dimensional RWRE X = (Z l, Z r ) reurns o zero infiniely ofen.

13 Muli-paricle processes wih reinforcemens 13 The following is equivalen saemen ha we can apply in our case: Suppose Z r is a RWRE on Z + such ha for any i 0, Z r jumps from sie i o i + 1 wih rae p 1 (i) and o i 1 wih rae 1 p 1 (i), and suppose Z l is a RWRE on Z such ha for any i 0, Z l jumps from sie i o (i + 1) wih rae p 2 (i) and o (i 1) wih rae 1 p 2 (i). If condiioned on he environmens {p 1 (i)} i and {p 2 (j)} j, Z r and Z l are independen birh and deah chains, hen Z r Z l is recurren. The proof of he above lemma can be hough of as an exercise on use of harmonic funcions in sochasic processes. I can be done wih Lyapunov funcions (see [6]), or alernaively wih conduciviies. 3.4 Proof of Theorem 1 I was essenial for proving Theorem 2 ha η = (l, r ) was defined via urns and magic marbles for τ 1. There are many ways o complee he proof of Theorem 1, one is o noice ha when he paricles separae afer he firs meeing ime τ 1, we can do he whole coupling consrucion (ha lead us o he proof of Theorem 2) anew, saring from scrach. The only hing differen will be he iniial marble configuraion for η = (l, r ), and he environmens of l P olya and r P olya, bu only a finiely many sies, hus esablishing he finieness of he second meeing ime τ 2. The finieness of τ 3, τ 4,... follows by inducion. This proof followed from he unusual coupling consrucion wih he magic marbles and he dominaion by wo RWREs, one on he righ and one on he lef. As we already menioned, he above dominaion can be consruced wihou using he magic marble approach, and generalized o work for more paricles han wo. However he approach aken in his paper allows us beer undersand he dynamics behind he wo-poin processes, and is valuable as an innovaive coupling echnique. Acknowledgmen The auhor wishes o hank all members of UCLA probabiliy group for heir suppor in he hree years ha he auhor worked here as a posdoc, and Silke Rolles who worked a UCLA a ha ime for useful discussions. While working on his version of he paper, he auhor received a lo of encouragemen from members of he probabiliy group a Oregon Sae Universiy. The auhor wishes o hank Rober Buron and Mina Ossiander for sharing houghs on nonexchangeable processes. References [1] R. M. Buron and G.Keller, Saionary measures and randomly chosen maps. Journal of Theoreical Probabiliy 6(1) (1993). [2] D.Coppersmih and P.Diaconis, Random walk wih reinforcemen. Unpublished manuscrip (1986)

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