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Localization of a vertex reinforced random walk on \(\mathbb{Z }\) with sub-linear weight

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We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number \(N\) larger than or equal to \(5\), there exists a vertex reinforced random walk which localizes with positive probability on exactly \(N\) consecutive sites.

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Correspondence to Bruno Schapira.

Appendix: proof of Proposition 1.4

Appendix: proof of Proposition 1.4

The proof of Proposition 1.4 is largely independent of the rest of the paper and uses arguments similar to those developed in [11, 12] and then in [1]. First, let us remark that the first part of the proposition is a direct consequence of Theorem \(1.1\) of [1]. Thus, we just prove (ii). Assume that localization on \(5\) sites occurs with positive probability and let us prove that necessarily \(i_-(w) = 2\). From now on, let \(\bar{X}\) denote the VRRW restricted to \([\![0, 4 ]\!]\) (i.e. reflected at sites \(0\) and \(4\)). Then, Lemma 3.7 of [1] insures that there exists some initial state \(\mathcal C \) such that \(\mathbb{P }_\mathcal{C }\{ \mathcal H \}>0\), where the event \(\mathcal H \) is defined by

$$\begin{aligned} \mathcal H :=\{\lim _{n\rightarrow \infty }\bar{Y}_n^+(0)<\infty \}\cap \{\lim _{n\rightarrow \infty }\bar{Y}_n^-(4)<\infty \} \end{aligned}$$


$$\begin{aligned} \bar{Y}_n^{\pm }(x) := \sum _{k=0}^{n-1} \frac{\mathbf{1}_{\{\bar{X}_k=x\;\mathrm{and }\;\bar{X}_{k+1}=x\pm 1\}}}{w(\bar{Z}_k(x\pm 1))}\qquad \text{ for } x\in \mathbb{Z }. \end{aligned}$$

Setting \(\bar{M}_n(x) := \bar{Y}_n^+(x)-\bar{Y}_n^-(x)\), we have, for any \(x\),

$$\begin{aligned} W(\bar{Z}_{n}(x+2))-W(\bar{Z}_{n}(x))=\bar{Y}_n^-(x+3)-\bar{Y}_n^+(x-1)+\bar{M}_{n}(x+1)+ C(x),\nonumber \\ \end{aligned}$$

where \(C(x)\) is some constant depending only on \(x\) and the initial state \(\mathcal{C }\). Moreover, for \(x\in [\![1, 3 ]\!]\), the process \((\bar{M}_n(x),n\ge 0)\) is a martingale bounded in \(L^2\). Therefore, recalling the notation \(\equiv \) defined in the beginning of Sect. 4, the a.s. convergence of \(\bar{M}_n(2)\) gives

$$\begin{aligned} W(\bar{Z}_n(3))\equiv W(\bar{Z}_n(1))\qquad \text{ on }\, \mathcal{H }. \end{aligned}$$

Using Lemma 2.1 and the fact that \(\bar{Z}_n(3)+\bar{Z}_n(1)\sim n/2\), we deduce that

$$\begin{aligned} \bar{Z}_n(1)\sim \bar{Z}_n(3)\sim \frac{n}{4} \qquad \text{ on }\, \mathcal{H }. \end{aligned}$$

Besides, the convergence of the martingale \(\bar{M}_n(3)\) combined with the fact that \(\bar{X}\) is reflected at site 4 imply that

$$\begin{aligned} \bar{Y}_n^-(3)\equiv \bar{Y}_n^+(3)\equiv W(\bar{Z}_n(4)). \end{aligned}$$

Hence, taking \(x=0\) in (65), we get

$$\begin{aligned} W(\bar{Z}_n(2))\equiv W(\bar{Z}_n(0))+W(\bar{Z}_n(4)). \end{aligned}$$

Define \(I_n :=\min (\bar{Z}_n(0),\bar{Z}_n(4))\) and \(S_n :=\max (\bar{Z}_n(0),\bar{Z}_n(4))\). The previous equation gives

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{W(I_n)}{W(\bar{Z}_n(2))}\le \frac{1}{2} \quad \text{ and }\quad \limsup _{n\rightarrow \infty } \frac{W(S_n)}{W(\bar{Z}_n(2))}\le 1, \end{aligned}$$

which implies, in view of Lemma 2.1,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\, \frac{I_n}{\bar{Z}_n(2)}=0 \quad \text{ and }\quad \limsup _{n\rightarrow \infty }\, \frac{S_n}{\bar{Z}_n(2)}\le 1. \end{aligned}$$

Using that \(I_n+S_n+\bar{Z}_n(2)\sim n/2\), we get

$$\begin{aligned} \liminf _{n\rightarrow \infty }\, \frac{\bar{Z}_n(2)}{n/4}\ge 1. \end{aligned}$$

In particular, denoting \(K_n:=\max (\bar{Z}_n(1),\bar{Z}_n(3))\sim n/4\), we deduce that for any \(\delta >0\) and for \(n\) large enough,

$$\begin{aligned} \bar{Z}_n(2)\ge (1-\delta )K_n. \end{aligned}$$

On the other hand, Equation (66) shows that there exists a (random) constant \(\gamma \), such that for \(n\) large enough,

$$\begin{aligned} W(\bar{Z}_n(2))\le 2W(S_n)+\gamma . \end{aligned}$$

Hence, we find that

$$\begin{aligned} K_n\le \frac{1}{1-\delta }W^{-1}\left( 2W(S_n)+\gamma \right) . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \bar{Y}^{+}_\infty (0)+\bar{Y}_\infty ^{-(4)} \,&= \, \sum _{n=0}^{\infty }\frac{\mathbf{1}_{\{\bar{X}_n=0\}}}{w(\bar{Z}_n(1))}+\frac{\mathbf{1}_{\{\bar{X}_n=4\}}}{w(\bar{Z}_n(3))} \,\ge \, \sum _{n=0}^{\infty } \frac{\mathbf{1}_{\{\bar{X}_n\in \{0,4\}\}}}{w(K_n)}\\&\ge \, c\sum _{n=0}^{\infty } \frac{\mathbf{1}_{\{\bar{X}_n\in \{0,4\}\}}}{w\left( \frac{1}{1-\delta }W^{-1}\left( 2W(S_n)+\gamma \right) \right) }\\&\ge c^{\prime }\sum _{k=0}^{\infty } \frac{1}{w\left( W^{-1}\left( 2W(k)+\gamma \right) \right) }, \end{aligned}$$

for some constants \(c,c^{\prime } > 0\). Recalling that

$$\begin{aligned} \Phi _{\eta ,3}(x)=W^{-1}\left( \int \limits _0^x \frac{dt}{w(\eta W^{-1}(W(x)/\eta ))}\right) , \end{aligned}$$

we deduce that if \(Y^+_\infty (0)+Y_\infty ^-(4)\) is finite with positive probability, then \(\Phi _{\eta ,3}(x)\) is bounded for any \(\eta <1/2\). This means that \(i_-(w)=2\), which concludes the proof of the proposition.

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Basdevant, A., Schapira, B. & Singh, A. Localization of a vertex reinforced random walk on \(\mathbb{Z }\) with sub-linear weight. Probab. Theory Relat. Fields 159, 75–115 (2014). https://doi.org/10.1007/s00440-013-0502-3

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  • Self-interacting random walk
  • Reinforcement
  • Regular variation

Mathematics Subject Classification

  • 60K35
  • 60J17
  • 60J20