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A random polymer approach to the weak disorder phase of the vertex reinforced jump process

Quentin Berger, Alexandre Legrand, Rémy Poudevigne, Christophe Sabot

TL;DR

The paper analyzes the L^p integrability of a VRJP-related martingale through a beta-potential/random-Schrödinger operator framework, focusing on weak disorder. It establishes an $L^{1+\delta}$ bound on the half-space for $W>W_c({\mathbb H}_d)$ and, in dimension $d\ge 4$, proves $L^p$ bounds for all $p\ge 1$ above the slab critical point $W_c^{\mathrm{slab}}$, via a renewal/overbreak approach and a detailed comparison to a toy model. The work connects the recurrence/transience of VRJP to the martingale limit, extends monotonicity arguments (Poudevigne) to infinite graphs, and draws inspiration from directed-polymer results to conjecture equivalences of critical thresholds across geometries. Overall, the results advance understanding of the weak disorder phase for VRJP, offering rigorous moment bounds and suggesting deep links between half-space, slab, and full-space behaviors with implications for the VRJP's diffusion properties.

Abstract

In this paper, we study the transient phase of the Vertex Reinforced Jump Process (VRJP) in dimension $d\geq 3$. In Sabot, Zeng (2019), the authors introduce a positive martingale and show that the VRJP is recurrent if and only if that martingale converges to $0$. On $\mathbb{Z}^d$, $d\ge 3$, with constant conductances $W$, it can be shown that there is a critical value $0<W_c(\mathbb{Z}^d)<\infty$, such that the martingale converges to $0$ if $W<W_c(\mathbb{Z}^d)$ or to a positive limit if $W>W_c(\mathbb{Z}^d)$. On the other hand, the VRJP martingale can be interpreted as the partition function of a non-directed polymer with a very specific $1$-dependent random potential. In this paper, we focus on the question of the $L^p$ integrability of the VRJP martingale, which is related to the (diffusive) behavior of the VRJP. First, taking inspiration from the work of Junk (2022) for directed polymers in $\mathbb{Z}^{1+d}$, we prove that on the half-space $\mathbb{H}_d$ of $\mathbb{Z}^d$, for all $W>W_c(\mathbb{H}_d)$ there is some $δ>0$ such that the VRJP martingale is in $L^{1+δ}$. Second, we prove that, in dimension $d\geq 4$, the VRJP martingale is in $L^{p}$ for all $p>1$ above the ``slab critical point'' $W_c^{\mathrm{slab}} (\mathbb{Z}^d) = \lim_{m\to\infty} W_c(\mathbb{Z}^{d-1} \times \{-m,\ldots,m\})$. We also propose some related conjectures.

A random polymer approach to the weak disorder phase of the vertex reinforced jump process

TL;DR

The paper analyzes the L^p integrability of a VRJP-related martingale through a beta-potential/random-Schrödinger operator framework, focusing on weak disorder. It establishes an bound on the half-space for and, in dimension , proves bounds for all above the slab critical point , via a renewal/overbreak approach and a detailed comparison to a toy model. The work connects the recurrence/transience of VRJP to the martingale limit, extends monotonicity arguments (Poudevigne) to infinite graphs, and draws inspiration from directed-polymer results to conjecture equivalences of critical thresholds across geometries. Overall, the results advance understanding of the weak disorder phase for VRJP, offering rigorous moment bounds and suggesting deep links between half-space, slab, and full-space behaviors with implications for the VRJP's diffusion properties.

Abstract

In this paper, we study the transient phase of the Vertex Reinforced Jump Process (VRJP) in dimension . In Sabot, Zeng (2019), the authors introduce a positive martingale and show that the VRJP is recurrent if and only if that martingale converges to . On , , with constant conductances , it can be shown that there is a critical value , such that the martingale converges to if or to a positive limit if . On the other hand, the VRJP martingale can be interpreted as the partition function of a non-directed polymer with a very specific -dependent random potential. In this paper, we focus on the question of the integrability of the VRJP martingale, which is related to the (diffusive) behavior of the VRJP. First, taking inspiration from the work of Junk (2022) for directed polymers in , we prove that on the half-space of , for all there is some such that the VRJP martingale is in . Second, we prove that, in dimension , the VRJP martingale is in for all above the ``slab critical point'' . We also propose some related conjectures.

Paper Structure

This paper contains 34 sections, 20 theorems, 149 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of the VRJP
  3. The random Schrödinger operator associated with the VRJP
  4. The martingale and polymer approach
  5. Phase transition: weak vs. strong disorder
  6. Statement of the results
  7. Main results I: moments of order larger than 1 on the half space
  8. Main results II: moments of all orders above the slab critical point
  9. Some related conjectures
  10. Comparison with the directed polymer model
  11. Organisation of the rest of the paper
  12. Preliminary material and complements concerning the beta-random potential
  13. The beta-potential on finite sets: definition, restriction and conditioning
  14. Extension of the beta-potential to infinite sets
  15. Convex Monotonicity
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $d\geq 3$. For any $W$ such that $\lim_{n\to\infty} M_n = M_{\infty}>0$ a.s., in particular for any $W>W_c$, there is some $\delta>0$ such that the convergence holds in $L^{1+\delta}$. In other words, we have this new characterization for $W_c$: The same result holds if we rather consider $M_n$ as the partition function of polymers on sequence of increasing finite boxes $(V_n)_{n\ge 0}$ such

Figures (4)

  • Figure 1: Illustration of the decomposition \ref{['eq:proof:claim1:1']}: we decompose the sum over paths in \ref{['eq:def:checkM:2']} according to whether the path stays inside $\Lambda_n$ before reaching $\partial_{k+1}^+$ (red/right path in the figure, first contribution in \ref{['eq:proof:claim1:1']}) or whether it first reaches $\partial^+_k\setminus \Lambda_n$ (blue/left path in the figure, second contribution in \ref{['eq:proof:claim1:1']}).
  • Figure 2: Illustration for the simplified graph $\widetilde{{\mathcal{G}} }$: in this picture $m=1$. The graph has vertex set $\llbracket -n,n\rrbracket\cup\{\star\}$. Each $i\in\llbracket -n,n\rrbracket$ is connected with its neighbours in $\llbracket -n,n\rrbracket$ by conductances $\varepsilon$. Moreover, the cemetery $\star$ is connected (represented with dashed lines) to the endpoints $-n$ and $n$ by conductances $\varepsilon$, and to all vertices $i\in V\cap (2m+1){\mathbb Z}$ which are not too close to the endpoints (i.e. $|i|\le n-m-2$) by i.i.d. conductances $\widetilde{W}_i\sim \mu_0$ (in blue).
  • Figure 3: Illustration of the different graphs used in the comparison: in each figure the green dots are identified with the cemetery $\star$ of the graph; the larger black dot represents the starting point of the VRJP. Step 0. The original graph on which $\psi_n$ is defined is denoted ${\mathcal{G}} _0$, with vertex set $V_n=\llbracket-n,n\rrbracket^d$; all inner boundary vertices are connected to the cemetery $\star$, see (A). Step 1. The graph ${\mathcal{G}} _1$ is obtained by removing edges so that the remaining ones form parallel slabs of width $2m+1$, connected along the vertical line $\{0\}^{d-1}\times \llbracket-n,n\rrbracket$, see (B). Step 2. The graph ${\mathcal{G}} _2$ (not represented) is defined by duplicating the vertical line in ${\mathcal{G}} _1$. Step 3. The graph ${\mathcal{G}} _3$ is obtained by removing some edges and lowering the weights: its vertical line has conductances $\varepsilon$, edges in the slabs have conductance $W_{c}^{(m)}+\varepsilon$, the center of each slab is connect to the vertical line through an edge with conductance $1$, see (C). Step 4. The graph ${\mathcal{G}} _3$ is equivalent to a realization of the simple graph $\widetilde{{\mathcal{G}} }$ defined in Definition \ref{['def:simplegraph']}, with blue edges having i.i.d. weight distribution $\mu_{n}^{m,\varepsilon}$.
  • Figure 4: Illustration of the graph $\widetilde{{\mathcal{G}} }_{\ell}$, here with $\ell = 9$.

Theorems & Definitions (50)

  • Remark 1.1
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Conjecture 2.4
  • Conjecture 2.5
  • Definition 1
  • Remark 3.1
  • Lemma 2
  • Definition 3
  • ...and 40 more