Vertex-reinforced random walk on ℤ eventually gets stuck on five points

Abstract

Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice Z. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.In other words, moves are restricted to the edges of G, with the probability of a move to a neighbor x being proportional to the augmented occupation Z n (x) of x at that time.VRRWs were introduced in 1988 by Pemantle [7] in the spirit of the seminal work by Coppersmith and Diaconis [4], who defined the notion of edge-reinforced random walks, which have at each step a probability to move along an edge proportional to the number of times plus 1 that the process has visited this edge. Reinforced processes are useful in models involving self-organization and learning behavior; they can also describe spatial monopolistic competition in economics. For more details on applications and known results in connection with these models, refer to the articles by Pemantle and Volkov [8,9].VRRWs on finite complete graphs, with reinforcements weighted by factors associated to each edge of the graph, have been studied by Pemantle [8] and Benam [1]. Pemantle and Volkov obtained results in 1997 on reinforced random walks on Z [9], which are described in the following text. More recently, Volkov [13] generalized some of these results and proved that, on a fairly broad class of locally finite graphs (containing the graphs of bounded degree), the VRRW has finite range with positive probability. The remainder of this paper is devoted to VRRWs on Z.Define the two random sets R := {v ∈ Z/ ∃ n ∈ N s.t. X n = v}, R ′ := {v ∈ Z/X n = v infinitely often} and, given k ∈ Z and α ∈ (0, 1), define the six events:Let | · | be the cardinality of a set. Pemantle and Volkov [9] proved the following results.Theorem 1.1. One has P(|R| < ∞) = 1 and P(|R| = 5) > 0.Theorem 1.2. One has P(|R ′ | ≤ 4) = 0. Theorem 1.3. For any open set I ⊂ [0, 1] and any integer k ∈ Z there exists, with positive probability, α ∈ I such that events 1-6 occur.Pemantle and Volkov also proposed the following conjecture.

Phase Transition in Vertex-Reinforced Random Walks on $${\mathbb{Z}}$$ with Non-linear Reinforcement

“…Part (b) immediately follows from Theorem 1.4 of Tarrès (10) , and part (c) from Theorem 3 and Remark 2. …”

Vertex-reinforced random walk on Z with sub-square-root weights is recurrent

“…Linearly vertex-reinforced random walk (VRRW for short), introduced by Pemantle [3], was studied on Z by Pemantle and Volkov [4]. A striking phenomenon was proved for this model [4,7]: the random walk will eventually visit just 5 sites on Z almost surely.…”

Martingale defocusing and transience of a self-interacting random walk

“…For instance the edge or vertex reinforced random walks have attracted a lot of attention, see e.g. [2,4,9,13,14,17,18,19]. Theorem 1.1.…”