Almost sure behavior of linearly edge-reinforced random walks on the half-line

We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^{\alpha} \vee 1$, and each time an edge is traversed, its weight is increased by $\Delta$. It is known that the walk is recurrent if and only if $\alpha \leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $\alpha<1$ and $\Delta>0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $\Delta>0$ is much slower than $\Delta=0$. In the critical case $\alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $\Delta=2$.


Introduction
Reinforced random walks (RRWs), introduced by Coppersmith and Diaconis, are a class of self-interacting random walks that have attracted many researchers for three decades or more. Quoting from Diaconis [8], "It was introduced as a simple model of exploring a new city. At first all routes are equally unfamiliar, and one chooses at random between them. As time goes on, routes that have been traveled more in the past are more likely to be traveled." Consider a finite connected graph, and each edge is given a positive initial weights. In each step the traveller jumps to an adjacent vertex by traversing an edge, with probability proportional to the weight of that edge. Each time an edge is traversed, its weight is increased by a fixed constant ∆ > 0 (linear edge-reinforcement). We can see that the walk is recurrent, that is, every vertex is visited infinitely often with probability one. The limiting density of the normalized occupation measure on the edges, obtained by Coppersmith and Diaconis in 1986, is found in [8] (see also Keane and Rolles [13]).
In this paper we consider the linearly edge-reinforced random walk (LERRW) on the half-line in recurrent regime, and give almost sure results on how far the traveller is from the origin. We begin with a motivating example. Let {S n } be the symmetric simple random walk on Z, starting at the origin. Notice that {|S n |} is the symmetric simple random walk on Z + = {0, 1, 2, · · · }, Now consider the LERRW on Z + , where the initial weights are all one and ∆ = 1. Let X n be the position of the walk at time n, and assume that X 0 = 0. Again {X n } is recurrent a.s. (see [4]), but is quantitatively quite different from {|S n |}: Theorem 2.2 below implies that lim sup n→∞ X n log 4 n = 1 a.s..
In this way, the random walk with reinforcement is much slower than ordinary random walk. We also discuss how strongly the linear reinforcement affects the long time behavior of the walk on Z + with heterogeneous initial weights, where each edge {x, x + 1} has initial weight x α ∨ 1. To summarize our aim is to describe phase transitions of the speed in the recurrent regime α ≤ 1 and ∆ ≥ 0. For α < 1, we obtain a limit theorem which shows strong slow-down effects from ∆ = 0, and is a counterpart of the law of the iterated logarithm for simple random walks. To our best knowledge this kind of results are first for RRWs. On the other hand, for α = 1, which is critical for recurrence, essential slow-down effects appear only for ∆ > 2.

2.1.
Model. We define the edge-reinforced random walk (ERRW) on Z + , denoted by X = {X n }, as follows. This process takes values on the vertices of Z + = {0, 1, 2, . . .}, and at each step it jumps to one of the nearest neighbors. For each x ∈ Z + , let f x = (f (ℓ, x) : ℓ ∈ Z + ) be a non-decreasing sequence of positive numbers, called the reinforcement scheme at x. For x ∈ Z + , let φ n (x) be the number of traversals of the edge {x, x + 1} by time n, namely For each n ≥ 0, the weights at time n are defined by We set w n (−1) = 0 for all n, which implies a reflection at the origin. We call w 0 (x) = f (0, x) the initial weight of the edge {x, x + 1}. Assume that P (X 0 = 0) = 1. The transition probability is given by .
The linearly edge-reinforced random walk (LERRW) is the ERRW whose reinforcement scheme is defined by We call ∆ ≥ 0 the reinforcement parameter.

2.2.
Recurrence classification. We say that the path X is recurrent if every point is visited infinitely often, and transient if every point is visited only finitely many times. The recurrence problem for LEERW on Z + is solved by Takeshima [22] (although only the ∆ = 1 case is treated in [22], his argument works for any ∆ > 0 as well). In Appendix A, we give an elementary and short proof of Theorem 2.1. .
Theorem 2.1 shows that the recurrence of the LERRW is completely determined by the initial weights: In particular, if the walk is transient when ∆ = 0, then it never becomes recurrent even if ∆ > 0 is very large.
Our first result is for the off-critical case, α < 1, and the precise order of oscillation of X n is indeed (log n) 1/(1−α) . Theorem 2.2. Assume that α < 1 and ∆ > 0. Let
Theorem 2.4. Assume that α = 1 and ∆ > 0, and consider the LERRW X with the initial weight (2.4) and the reinforcement parameter ∆.

Effect of linear reinforcement.
For comparison, we give almost sure bounds for unreinforced case. In the case α < 1 the speed of the walker becomes much slower as soon as ∆ > 0, while it is not in the critical case α = 1.

Related works.
We briefly review related literatures concerning limit theorems for ERRWs in one dimension. In Davis [4], the strong law of large numbers lim n→∞ X n n = 0 a.s.
is proved for initially fair, sequence-type RRWs (that is, f x does not depend on x). See also Takeshima [22] for a possible generalization. For limit theorems for sublinear ERRWs, see [5,24,25] among others. The continuous time vertex-reinforced jump process (VRJP) was introduced by Davis and Volkov [7]. The LERRW and the VRJP are known to be closely related, see Sabot and Tarrès [20] and references therein. The analog of Theorem 2.1 for VRJP on Z + is proved in Davis and Dean [6]. For the VRJP {X t } on Z + corresponding to the LERRW with f (x, ℓ) = 1 + ℓ, Davis and Volkov [7] shows that lim t→∞ 1 log t max 0≤s≤t X s = 2.768 · · · a.s..
In Lupu, Sabot, and Tarrés [15], the continuous space limit of the VRJP in one dimension is constructed, and it is also obtained as a fine mesh limit of the LERRW.

Preliminaries
3.1. Reduction of LERRW to RWRE. Following Pemantle [19], we introduce a random walk in random environment (RWRE), which is equivalent to the LERRW on Z + with ∆ > 0.
The expectation and variance under P are denoted by E[ · ] and V[ · ], respectively. Given a random environment for n ≥ 0 and i ∈ Z + . The next result is found in [19], Section 3. (See also Eckhoff and Rolles [9] for the uniqueness of representation.) Lemma 3.1. For any n ≥ 0 and any i 0 , i 1 , · · · , i n ∈ Z + , we have RW in a fixed environment. In this subsection, we fix an environment {p i }.
Define {γ x } x∈Z + by In the electric network interpretation (see e.g. Chapter 2 in Lyons and Peres Using the conductance w x := 1/γ x of the edge {x, x + 1}, we have where w −1 := 0. Define {π x } x∈Z + by From (3.1), we can see that {π x } is a reversible measure. Notice that The following recurrence classification is classical (see e.g. Theorem 2.2.5 in [18]).

Lemma 3.2. Consider the random walk
The unique stationary distribution is given by 4. Almost sure bound 4.1. Almost sure bound by the Lyapunov function method. We consider the RWRE Y , defined in the previous section. The first hitting time to x ∈ Z + is defined by The next lemma is a consequence of the hitting time identity (see Proposition 2.20 in [16]).
for x ∈ Z + . Then the expectation of τ x under P ω 0 is given by . To obtain the almost sure upper bound, we use the following lemma (see Lemma 6.1.4 and Theorem 2.8.1 in [18]). Lemma 4.2. Let t 1 be an increasing, nonnegative function on Z + with t 1 (x) → ∞ as x → ∞. If P-a.e. ω, T ω (x) ≥ t 1 (x) for all but finitely many x ∈ Z + , then for any ε > 0, P-a.e. ω and P ω 0 -a.s., for all but finitely many n.
As for the almost sure lower bound, we use the following version of Lemma 4.3 in [10]. No essential change is needed for the proof.

Lemma 4.4.
For any x ∈ Z + , we have π i < ∞, then (4.3) can be improved as follows: 4.2. LERRW with ∆ = 0. As a warm-up, we prove almost sure bounds for the case ∆ = 0. We use the next lemma, which is an infinite series version of l'Hospital's rule, due to Stolz and Cesàro. for T ω (x).

By Lemmata 4.2 and 4.3, we have
for all large n, for infinitely many n, By Lemma 4.5, For simplicity, we content ourselves with a weaker bound: For any ε > 0, x 2+ε for all but finitely many x.
We can obtain the conclusion of (ii) by a similar calculation as in (i).
(iii) Suppose that −1 < α ≤ 1. We have and again by Lemma 4.5. The rest of the proof is the same as above.

Proof of main theorems
The following proposition allows us to estimate the random resistance {γ x } x∈Z + .
Proposition 5.1. Assume that {p i (ω)} i∈N is a sequence of independent random variables, and the distribution of p i is Beta (i) If α < 1 and ∆ > 0, then The proof of Proposition 5.1 consists of several steps, and will be given in the next section. We prove our main results first. Notice that Z ω = ∞ x=0 π x < +∞ if α < 1 and ∆ > 0, or α = 1 and ∆ > 2.
6. Proof of Proposition 5.1 We begin with a particularly simple case, α = 0 and ∆ > 0. Since {p i } i∈N is an i.i.d. sequence with , the strong law of large numbers for i.i.d. sequences together with (4.1) in [22] imply that If ∆ = 1, then we have To obtain the result for the other cases, we prepare some lemmata. As (4.15) and (4.13) in [22], for x ∈ N,
The following lemma is a consequence of Kolmogorov's strong law of large numbers (see e.g. [12], Theorem 4.5.2).
Letting k → ∞, we obtain (A. If F 0 = +∞, then P (E) cannot be positive. The conclusion follows from Lemma A.1.