A random environment for linearly edge-reinforced random walks on infinite graphs

Abstract

We consider linearly edge-reinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a random walk in a time-independent random environment given by strictly positive weights on the edges. Furthermore, we prove bounds for the random environment, uniform, among others, in the size of the graph. 34

Edge-reinforced random walk on one-dimensional periodic graphs

“…Although (X t ) t∈N 0 is not a Markov chain, it was shown in [8] that the edge-reinforced random walk is recurrent if and only if it returns to the starting point at least once with probability one.…”

“…In [8], it is shown that the edge-reinforced random walk on any graph with arbitrary initial weights has the same distribution as a mixture of reversible Markov chains. For the periodic graphs defined above with periodic initial weights, we can say more: Denote by := (0, ∞) E the set of strictly positive edge weights for the graph G. For x = (x e ) e∈E ∈ , set…”

“…Since the graph G (N ) is periodic and connected, there is a constant L such that for all N and all vertices w, w ∈ V (N ) with level(w) = level(w ) there exists a path from w to w in G(N ) of length at most L. Hence, Theorem 2.4 of[8] implies that there exist constants c11 = c 11 (a, L) > 0 and c 12 = c 12 (a, L) > 0, depending only on a (but not on N ) such that for all N , all vertices w, w ∈ V (N ) with level(w) = level(w ), all edges e w, e w , and all M > 0, one has Q (N ) 0 [x e ≥ M x e ] ≤ c 11 M −c 12 . (4.62)…”

“…Thus, the current approach needs only a few existing results about the edge-reinforced random walk: The explicit form of the distribution of the random environment is used, but only some simple calculations have to be performed with it. Furthermore, we use a tail estimate from [8] for quotients of the random weights constituting the random environment.…”

“…(5.2) Note that by definition the graphG (m) = (Ṽ(m) ,Ẽ(m) ) is connected, that the absolute value of the level of all vertices inṼ (m) is bounded by Rm, and thatṼ (m) ↑ V as m → ∞.For N large enough, more specifically for N ≥ 2(m + 1)R, the canonical map mod N :Ṽ (m) → V (N ) , (i, u) → (i mod N , u) yields a graph isomorphism ofG (m)to a full subgraph of G (N ) ; we use this graph isomorphism to identifyG (m) with its image. Now fix m ∈ N and let N ≥ 2(m + 1)R. Theorem 2.4 of[8], applied to the graph G (N ) , yields for all e ∈Ẽ (m) and all M > 0 the boundsQ (N ) 0 x e 0 ≥ M x e ≤ c 15 m M −c 16 /m , Q (N ) 0 x e ≥ M x e 0 ≤ c 15 m M −c 16 /m (5.3)with some positive constants c 15 = c 15 (a) and c 16 = c 16 (a) = min{a e /2 : e ∈ E} depending only on the initial weights a, but not on N ; recall that e 0 denotes a reference edge adjacent to 0, and note that e 0 and e are connected by a path of at most m intermediate vertices. In particular, for any fixed m, the laws Q(N ) 0 [(log(x e /x e 0 )) e∈Ẽ (m) ∈ ·], N ≥ 2(m + 1)R, are tight.…”