Monotonicity for excited random walk in high dimensions

Abstract

We prove that the drift θ(d, β) for excited random walk in dimension d is monotone in the excitement parameter β ∈ [0, 1], when d is sufficiently large. We give an explicit criterion for monotonicity involving random walk Green's functions, and use rigorous numerical upper bounds provided by Hara (Private communication, 2007) to verify the criterion for d ≥ 9.

Limit laws of transient excited random walks on integers

“…If the site had been visited before, then the walk made unbiased jumps to one of its nearest neighbor sites. See [17], [4], [13] and references therein for further results about this particular model.…”

On strict monotonicity of the speed for excited random walks in one dimension

“…the transition probabilities do not depend on the site of departure) the environment is said to be non-random. The original model, introduced by Benjamini and Wilson [5], where β(x, 1, e) = (1+ βe⋅e 1 ) (2d) and β(x, k, e) = 1 (2d) for all e when k > 1, has been extensively studied, and in particular it is known that the limiting velocity of the walker lim n→∞ n −1 X n = v(β) is (deterministic and) strictly positive when d > 1 [6,16,15] and strictly monotone in β when d > 8 [10] (see also [19,8,12]). It is believed that this monotonicity holds in dimensions d > 1, but the absence of any natural coupling argument has been an obstacle to resolving this conjecture.…”

A combinatorial result with applications to self-interacting random walks

“…A natural first attempt at trying to prove such a monotonicity result would be as follows: given 0 < β 1 < β 2 ≤ 1, construct a coupling of excited random walks X and Y with parameters β 1 and β 2 > β 1 respectively such that with probability 1, X [1] n ≤ Y [1] n for all n. Thus far no one has been able to construct such a coupling, and the monotonicity of v [1] β as a function of β remains an open problem in dimensions 2 ≤ d ≤ 8. In dimensions d ≥ 9 this result has been proved [3] using a somewhat technical expansion method, as well as rigorous numerical bounds on simple random walk quantities. More general models in 1 dimension have been studied, and some monotonicity results [6] have been obtained via probabilistic arguments but without coupling.…”