Dayue Chen Outline Joint work with Zhichao Shan, submitted to SPA. - PowerPoint PPT Presentation

The Voter Model in a Random Environment in Z d Dayue Chen Peking University December 5, 2011, Kochi, Japan The Voter Model in a Random Environment in Z d Dayue Chen

The Voter Model in a Random Environment in Z d Dayue Chen

Outline Joint work with Zhichao Shan, submitted to SPA. 1. a new result of the voter model 2. collision of two random walks The Voter Model in a Random Environment in Z d Dayue Chen

Introduction: the model The voter model is an interacting particle system. There is a voter in every site of V . Every voter can have either of two political positions, denoted by 0 or 1, and constantly updates his political position. The voter at x updates his political position at a random time, following the exponential distribution with parameter � z µ xz . At the time of update the voter takes the position of his neighbor y with probability µ xy / ( � z µ xz ). Let η ( x ) be the political position of voter x and the collection η = { η ( x ); x ∈ V } be an element of { 0 , 1 } V . The Voter Model in a Random Environment in Z d Dayue Chen

Introduction: construction The voter model can be constructed either by the Markovian semigroup or by the graphical representation, see Liggett(85). The second approach not only works for all positive µ xy , but also clearly exhibits the duality relation. The Voter Model in a Random Environment in Z d Dayue Chen

Introduction: limit behavior When the underlying graph is Z d and µ e ≡ 1, this model is well studied. There are two invariant measures δ 0 and δ 1 , and if d ≤ 2, all other invariant measures are linear combinations of δ 0 and δ 1 . The Voter Model in a Random Environment in Z d Dayue Chen

New Result The underlying graph is Z d and { µ e , e ∈ E d } are i.i.d. random variables satisfying µ e ≥ 1. The measures δ 0 and δ 1 of point mass are invariant. Theorem Let d = 1 or 2. Suppose that ( µ e ) are i.i.d. and µ e ≥ 1 P -a.s. There exists Ω 0 ⊆ Ω with P (Ω 0 ) = 1 . For any ω ∈ Ω 0 , the voter model has only two extremal invariant measures: δ 0 and δ 1 . Remark: I. Ferreira, The probability of survival for the biased voter model in a random environment, Stochastic Processes and Their Appl. , vol.34, (1990), 25–38. I. Ferreira, Cluster for the Voter Model in a Random Environment and the probability of survival for the Biased Voter Model in a Random Environment, 1988 The Voter Model in a Random Environment in Z d Dayue Chen

Duality Relation For η ∈ { 0 , 1 } Z 2 and a finite set A ⊆ Z 2 , define H ( η, A ) = 1 { η ( z )=1 for all z ∈ A } . If there are two Markov processes, { η t } and { A t } , such that E η ω H ( η t , A ) = E A ω H ( η, A t ) , Then we say { η t } and { A t } are dual to one another. The Voter Model in a Random Environment in Z d Dayue Chen

Coalescing Random Walk can be a dual of the voter model. taking values on the set of all finite sets of vertices of Z d . Intuitively, image there is a particle at each x ∈ A of the initial state. Each particle performs a variable speed random walk, independent of each other until they meet. Once two particles collide, they coalesce into one particle. Then A t is the set of locations of all particles at time t . { A t } and the voter model can be constructed by the same graphical representation. P η ω ( η t ( x ) = 1 for all x ∈ A ) = P A ω ( η ( x ) = 1 for all x ∈ A t ) . The Voter Model in a Random Environment in Z d Dayue Chen

Reducing to the collision problem If the initial state is a singleton and if singleton { x } is identified with vertex x , then the coalescing Markov chain is exactly a continuous-time random walk in a random environment (or variable speed random walk or the random conductance model ). Theorem Let d = 2 . Suppose that ( µ e , e ∈ E d ) are i.i.d. and µ e ≥ 1 P -a.s. There exists Ω 0 ⊆ Ω with P (Ω 0 ) = 1 . Let ω ∈ Ω 0 and P ω denote the probability conditional on the environment. If { X t } and { Y t } are two independent variable speed random walks starting from x and y respectively, then P ω ( X t = Y t for some t ≥ 1) = 1 . = ⇒ Starting from a doubleton (or a finite set), a coalescing Markov chain will eventually becomes a singleton. = ⇒ Any invariant measure of the voter model is a linear combinations of δ 0 and δ 1 . The Voter Model in a Random Environment in Z d Dayue Chen

Collisions of Random Walks The dual relation lead Liggett in 1974 to first consider collisions of two Markov chains, and to discover an example that two recurrent Markov chain may not necessarily meet each other. Krishnapur and Peres (2004) found a simple example. A recent paper by Barlow, Peres and Sousi. Xinxing Chen and I also made contributions. Many progresses, yet some questions remain open. The Voter Model in a Random Environment in Z d Dayue Chen

Part II Collisions of Random Walks in a random environment The Voter Model in a Random Environment in Z d Dayue Chen

The Proof: The Main Lemma Theorem Let d = 2 . Suppose that ( µ e , e ∈ E d ) are i.i.d. and µ e ≥ 1 P -a.s. There exists Ω 0 ⊆ Ω with P (Ω 0 ) = 1 . Let ω ∈ Ω 0 and P ω denote the probability conditional on the environment. If { X t } and { Y t } are two independent variable speed random walks starting from x and y respectively, then P ω ( X t = Y t for some t ≥ 1) = 1 . can be deduced by the 2nd Borel-Cantelli Lemma from Lemma Under the same assumption, P ω ( X t = Y t for some t ≥ 1) ≥ δ > 0 , where δ is a constant independent of ω , x and y. The Voter Model in a Random Environment in Z d Dayue Chen

The Proof: From Lemma to Theorem Let δ > 0 be defined as before. Fix ω ∈ Ω 0 . By Lemma 4, there exists a function f : V 2 × V 2 �→ [1 , ∞ ), such that for all x , y ∈ V 2 , ( X t = Y t for some 1 < t ≤ f ( x , y )) ≥ δ P ( x , y ) 2 . (1) ω Set x 0 = x , y 0 = y and t 0 = 0. Define x i , y i and t i inductively for i ≥ 1 as follows. Suppose that x i , y i and t i are already defined. Let { ˜ X t } and { ˜ Y t } be two independent continuous-time random walks starting from x i and y i . Define x i +1 := ˜ y i +1 := ˜ X ( f ( x i , y i )) , Y ( f ( x i , y i )) , and t i +1 := t i + f ( x i , y i ) . Define E i to be the event that X t = Y t for some t ∈ ( t i + 1 , t i +1 ] for i ≥ 0. By (1) and the strong Markov property, Y t for some 1 < t ≤ f ( x i , y i )) ≥ δ ( ˜ X t = ˜ P ω ( E i | X t , Y t , t ≤ t i ) = P ( x i , y i ) 2 . ω By the second Borel-Cantelli lemma, P ω ( E i infinitely often)=1. P ω ( X t = Y t infinitely often) ≥ P ω ( E i infinitely often) = 1 . The Voter Model in a Random Environment in Z d Dayue Chen

Proof of the Lemma based on another lemma Define the random variable � T 1 H := µ ( X s ) µ ( Y s )1 { X s = Y s ∈ M ( s 1 / 2 ) } d s . t 0 where t 0 and T are constants to be specified later, as well as the subset M ( n ). Lemma E ω H ≥ c 9 log T . E ω H 2 ≤ (4 π c 2 3 + 2 π 2 c 4 3 / c 4 )(log T ) 2 . P ω ( X t = Y t for some t > 0) ≥ P ω ( H > 0) ≥ ( E ω H ) 2 E ω H 2 ( c 9 log T ) 2 c 2 9 c 4 ≥ 3 / c 4 )(log T ) 2 = > 0 . (4 π c 2 3 + 2 π 2 c 4 4 π c 2 3 c 4 + 2 π 2 c 4 3 The Voter Model in a Random Environment in Z d Dayue Chen

Key Ingredient: Heat Kernel Est. by Barlow & Deuschel Theorem Let d ≥ 2 and σ ∈ (0 , 1) . There exist random variables S x , x ∈ Z d , such that P ( S x ( ω ) ≥ n ) ≤ c 1 exp( − c 2 n σ ) , (2) and constants c i (depending only on d and the distribution of µ e ) such that the following hold. If | x − y | 2 ∨ t ≥ S 2 x , then t ( x , y ) ≤ c 3 t − d / 2 e − c 4 | x − y | 2 / t when t ≥ | x − y | , q ω q ω t ( x , y ) ≤ c 3 exp( − c 4 | x − y | (1 ∨ log( | x − y | / t ))) when t ≤ | x − y | . If t ≥ S 2 x ∨ | x − y | 1+ σ , then t ( x , y ) ≥ c 5 t − d / 2 e − c 6 | x − y | 2 / t . q ω (3) The Voter Model in a Random Environment in Z d Dayue Chen

The Proof Lemma Let A n ( ω ) be the random set defined by A n ( ω ) = { x : | x | ≤ n , S x ( ω ) ≤ 2 log n } . Then almost surely there exists a finite random variable U ( ω ) such that | A n ( ω ) | ≥ c 7 n 2 for any n ≥ U ( ω ) . For any x , y ∈ Z 2 set t 0 = [ S x ( ω ) ∨ S y ( ω )] 2 + [ U ( ω ) + ( | x | ∨ | y | )(1 + 12 π c − 1 7 )] 2 , 2 and T = exp( 1+ σ log t 0 ), where σ is given in the previous theorem. B x ( r ) = disk of radius r centered at x , M ω ( n ) = B x ( n ) ∩ B y ( n ) ∩ A n ( ω ). for n ≥ U ( ω ) + ( | x | ∨ | y | )(1 + 12 π c − 1 | M ω ( n ) | > C 7 n 2 / 2 7 ). The Voter Model in a Random Environment in Z d Dayue Chen

The Proof: Lower bound of E ω H � T 1 E ω H = µ ( X s ) µ ( Y s ) 1 { X s = Y s ∈ M ( s 1 / 2 ) } d s E ω t 0 � T 1 � = P ω ( X s = z , Y s = z ) d s µ 2 t 0 z z ∈ M ( s 1 / 2 ) � T � q ω s ( x , z ) q ω = s ( y , z ) d s . t 0 z ∈ M ( s 1 / 2 ) Since z ∈ M ( s 1 / 2 ), we have | x − z | 2 ≤ s ≤ T = exp( 2 1+ σ log t 0 ). Thus s ≥ t 0 ≥ S 2 x ( ω ) ∨ | x − z | 1+ σ . Theorem s ( x , z ) ≥ c 5 s − d / 2 e − c 6 | x − z | 2 / s . If s ≥ S 2 x ∨ | x − z | 1+ σ , then q ω Similarly s ≥ S 2 y ( ω ) ∨ | y − z | 1+ σ . The Voter Model in a Random Environment in Z d Dayue Chen