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Recurrence of the VRJP and Exponential Decay in the \(H^{2|2}\)-Model on the Hierarchical Lattice for \(d\le 2\)

Jinglin Wang, Xiaolin Zeng

TL;DR

The paper analyzes the VRJP on a hierarchical lattice and its SUSY $H^{2|2}$-model, proving a phase transition: recurrence for $d\le 2$ and transience for $d>2$, with $d=2$ showing critical behavior governed by reinforcement. It develops a fractional-moment method tailored to the hierarchical structure, employing a coarse-graining lemma that exploits linear reinforcement to obtain exponential decay of the effective field $u$, and thereby local order in the $H^{2|2}$-model. The results are achieved via uniform finite-box bounds on the Green’s function and an effective-conductance/Rayleigh-monotonicity framework to connect decay rates to recurrence or transience. The work clarifies how entropy and the full connectivity of the hierarchical lattice are managed, yielding optimal decay rates and a robust connection between VRJP behavior and $H^{2|2}$-model correlations, especially around the critical dimension $d=2$.

Abstract

We show that the vertex-reinforced jump processes on a \(d\)-dimensional hierarchical lattice are recurrent for \(d < 2\) and transient for \(d > 2\). We also explore certain regimes when \(d = 2\). The proof of recurrence relies on an exponential decay estimate of the fractional moment of the Green's function, which, unlike the classical approach used for \(\mathbb{Z}^d\), requires additional entropy estimates via stability of the model distribution under coarse grain operation, which leverages its linear reinforcement.

Recurrence of the VRJP and Exponential Decay in the \(H^{2|2}\)-Model on the Hierarchical Lattice for \(d\le 2\)

TL;DR

The paper analyzes the VRJP on a hierarchical lattice and its SUSY -model, proving a phase transition: recurrence for and transience for , with showing critical behavior governed by reinforcement. It develops a fractional-moment method tailored to the hierarchical structure, employing a coarse-graining lemma that exploits linear reinforcement to obtain exponential decay of the effective field , and thereby local order in the -model. The results are achieved via uniform finite-box bounds on the Green’s function and an effective-conductance/Rayleigh-monotonicity framework to connect decay rates to recurrence or transience. The work clarifies how entropy and the full connectivity of the hierarchical lattice are managed, yielding optimal decay rates and a robust connection between VRJP behavior and -model correlations, especially around the critical dimension .

Abstract

We show that the vertex-reinforced jump processes on a -dimensional hierarchical lattice are recurrent for and transient for . We also explore certain regimes when . The proof of recurrence relies on an exponential decay estimate of the fractional moment of the Green's function, which, unlike the classical approach used for , requires additional entropy estimates via stability of the model distribution under coarse grain operation, which leverages its linear reinforcement.
Paper Structure (10 sections, 9 theorems, 104 equations, 4 figures)

This paper contains 10 sections, 9 theorems, 104 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of models and results
  3. Finite box approximations
  4. Results and strategies of proofs
  5. Recurrence and transience of the VRJP, proof of Theorem \ref{['thm-vrjp-phase-trans']}
  6. Recurrence of VRJP when $d\leq 2$
  7. Transience of VRJP when $d>2$
  8. Fractional moment estimates
  9. Estimate for $G(i,\delta)$
  10. Estimate for $G(i,j)$

Key Result

Lemma 1

Let $\mathcal{G}=(V,E,W)$ be an edge weighted finite (non oriented) graph with edge weight $W_{i,j}$, consider the random Schrödinger matrix $H_{\beta}$, s.t. $H_{\beta}(i,j)=-W_{i,j}$ if $i\ne j$ and $H_{\beta}(i,i)=2\beta_{i}$, the law of $H_{\beta}$ is defined via the following p.d.f. call such random matrix $H_{\beta}$ the random Schrödinger matrix of $H^{2|2}$-mode on $\mathcal{G}$. For any

Figures (4)

  • Figure 1: Plot of the empirical $1/4$-moment of $e^{u^{(1)}_{i}},0\leq i\leq n$ with sample size $50$ on $\Lambda_{10}$, i.e. graph with 1024 vertices, with parameter $W=1$ and different $\rho$, i.e. spectral dimensions $d=2 \frac{\log 2}{\log \rho}$. This plot is illustrative and not intended to be quantitative.
  • Figure 2: The box $\widetilde{\Lambda}_4$, note that $\delta_4$ is also $B_4$.
  • Figure 3: The graph $\widetilde{\Lambda}_{5,\{1,9\}}^{(0)}$ after coarse-grain in the box $\widetilde{\Lambda}_5$ to compute $G_{\widetilde{\Lambda}_5}(1,9)$, note that $\delta_5$ is also $B_5$.
  • Figure 4: Illustration of the inequality \ref{['eq-recursion-fmm-Gij']}, where fractional moment of $G(B_{k'},B_j)$ is bounded by sum over $G(B_{i_1},B_j)$ and $G(B_j,B_{i_2})$.

Theorems & Definitions (9)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Corollary 6
  • Proposition 7
  • Proposition 8
  • Proposition 9