General many-dimensional excited random walks Mikhail Menshikov - PowerPoint PPT Presentation

“Simple” ERW in dimension 2 Generalized ERW Proofs General many-dimensional excited random walks Mikhail Menshikov Serguei Popov Alejandro Ramirez Marina Vachkovskaia Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs “Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Excited Random Walk (ERW) was introduced by Benjamini and Wilson (2003). It is a discrete-time process that lives in Z d , and can be informally described as follows: ◮ fix a parameter p ∈ ( 1 2 , 1 ] ◮ if the walk is at a site x which was already visited, it jumps with probabilities 1 / ( 2 d ) to the nearest neighbor sites of x ◮ if the process visits a site x for the first time, it jumps to the right (i.e., in the direction of the first coordinate vector e 1 ) with probability p / d , to the left with probability ( 1 − p ) / d and to the other nearest neighbor sites of x with probability 1 / ( 2 d ) . Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Informal interpretation: ◮ initially each site contains one cookie ◮ the particle eats all cookies it finds ◮ immediately after eating a cookie, the particle gets a “bias” to the right ◮ no cookie = no bias Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Known results (Benjamini, Wilson, Kozma, Bérard, Ramirez, van ver Hofstad, Holmes), case d ≥ 2: ◮ ERW is transient to the right ◮ ERW is ballistic to the right ◮ LLN ◮ CLT ◮ monotonicity in p in high dimensions Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Proof (main ideas, d = 2): ◮ coupling of ERW X with SRW Y , such that ( X n − Y n ) · e 1 is nondecreasing in n and ( X n − Y n ) · e 2 = 0 ◮ tan points for SRW. A tan point in dimension d = 2 is defined as any site x ∈ Z 2 with the property that the ray { x + ke 1 : k ≥ 0 } is visited by the SRW for the first time at site x . Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs x y 0 0 x is a tan point, y is not a tan point Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs It is known (Bousquet-Mélou, Schaeffer (2002)) that with “large” probability, the number of tan points up to time n is at least 3 4 − ε . n Let R n be the set of sites visited up to time n . |R n | ≥ the number of tan points of Y by time n (using the coupling with SRW) So (taking ε < 1 4 ) 4 − ε ≫ n 3 1 |R n | > n 2 with “large” probability. Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Proof of transience: M n = X n − 2 p − 1 |R n | is a martingale. 2 Azuma inequality: if { Z n } n ∈ N is a martingale with respect to some filtration, and such that | Z k − Z k − 1 | < c a.s., then a 2 � � P [ | Z n − Z 0 | ≥ a ] ≤ 2 exp − . 2 nc 2 1 1 2 + δ ), since we should have |R n | ≫ n 2 , it holds So (take a = n 1 2 , and then one also that (again, with “large” probability) X n ≫ n can use Borel-Cantelli to obtain transience to the right. Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Proofs of LLN and CLT (here ℓ = e 1 ): τ 3 τ 1 τ 2 0 ℓ regeneration structure + estimates on tails of τ k + 1 − τ k Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs What if we modify the model? ◮ drift in cookies not parallel to e 1 ◮ different drifts in different cookies ◮ SRW → some RW with zero drift and bounded jumps ◮ etc. — there are difficulties, because we cannot use the coupling with SRW and tan points! Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs “Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Generalized ERW is a discrete-time process X in Z d , d ≥ 2, satisfying the following conditions: Condition B. There exists a constant K > 0 such that sup n ≥ 0 � X n + 1 − X n � ≤ K a.s. Condition E. Let ℓ ∈ S d − 1 . We say that Condition E is satisfied with respect to ℓ if there exist h , r > 0 such that for all n P [( X n + 1 − X n ) · ℓ > r | F n ] ≥ h and for all ℓ ′ with � ℓ ′ � = 1, on { E ( X n + 1 − X n | F n ) = 0 } P [( X n + 1 − X n ) · ℓ ′ > r | F n ] ≥ h . Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Condition C + . Let ℓ ∈ S d − 1 . We say that Condition C + is satisfied with respect to ℓ if there exist a λ > 0 such that E ( X n + 1 − X n | F n ) = 0 on {∃ k < n such that X k = X n } , and E ( X n + 1 − X n | F n ) · ℓ ≥ λ on { X k � = X n for all k < n } . Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Results: Theorem Let d ≥ 2 and ℓ ∈ S d − 1 . Assume that X is a generalized excited random walk in direction ℓ . Then, there exists v = v ( d , K , h , r , λ ) > 0 such that X n · ℓ lim inf ≥ v a.s. n n →∞ Also, for “homogeneous” ERW and ERW in i.i.d. random environment we prove LLN and (averaged) CLT. They follow from the regeneration times argument, using the estimates obtained in the course of the proof of the above result. Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs “Simple” ERW in dimension 2 Generalized ERW Proofs Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks

“Simple” ERW in dimension 2 Generalized ERW Proofs Define H ( a , b ) = { x ∈ Z d : x · ℓ ∈ [ a , b ] } , and n � L n ( m ) := 1 { X j · ℓ ∈ [ m , m + 1 ) } . j = 0 Lemma Let X ′ be a submartingale in direction ℓ with uniformly bounded jumps and uniform ellipticity. Then, for any δ > 0 there exists a constant γ ′ 1 such that for all m we have 1 1 n δ . 2 + 2 δ ] ≤ e − γ ′ P [ L n ( m ) ≥ n Menshikov, Popov, Ramirez, Vachkovskaia General many-dimensional excited random walks