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On Anti-Confinement Estimates for Self-Repelling Random Walks

Tobias Schmidt, Mark Sellke

TL;DR

The paper analyzes self-repelling self-interacting random walks in $d$ dimensions reweighted by a Gibbsian pair potential $W$, establishing diffusion-constant bounds for short-range interactions and identifying superdiffusive behavior for sufficiently long-range temporal decay. Central to the methodology are GKS correlation inequalities on path space and Wells’ reduction to one dimension, enabling a dyadic, multi-scale variance recursion that amplifies fluctuations across scales. A key technical contribution is a universal bound on tilted pairwise means under symmetric laws, reducing to a two-point extremal measure via convexity arguments, which underpins the recursive growth that yields superdiffusivity for $\xi<\gamma/2+c_{\mathrm{crit}}$. The results illuminate how temporal decay and spatial repulsion interplay to control diffusion, with implications for polymer-like Gibbs measures and related quantum-field–path representations.

Abstract

We study a class of $d$-dimensional random walks, including the two-dimensional simple random walk, reweighted by a self-repelling Gibbsian pair potential. We prove lower bounds on the diffusion constant for short-range interactions, and superdiffusive behavior in case the interaction is sufficiently long-range. Finally, we show that in the superdiffusive regime, faster temporal decay can be compensated by stronger spatial repulsion and vice-versa. Our technique combines GKS-based correlation inequalities on path space with recursive multi-scale estimates.

On Anti-Confinement Estimates for Self-Repelling Random Walks

TL;DR

The paper analyzes self-repelling self-interacting random walks in dimensions reweighted by a Gibbsian pair potential , establishing diffusion-constant bounds for short-range interactions and identifying superdiffusive behavior for sufficiently long-range temporal decay. Central to the methodology are GKS correlation inequalities on path space and Wells’ reduction to one dimension, enabling a dyadic, multi-scale variance recursion that amplifies fluctuations across scales. A key technical contribution is a universal bound on tilted pairwise means under symmetric laws, reducing to a two-point extremal measure via convexity arguments, which underpins the recursive growth that yields superdiffusivity for . The results illuminate how temporal decay and spatial repulsion interplay to control diffusion, with implications for polymer-like Gibbs measures and related quantum-field–path representations.

Abstract

We study a class of -dimensional random walks, including the two-dimensional simple random walk, reweighted by a self-repelling Gibbsian pair potential. We prove lower bounds on the diffusion constant for short-range interactions, and superdiffusive behavior in case the interaction is sufficiently long-range. Finally, we show that in the superdiffusive regime, faster temporal decay can be compensated by stronger spatial repulsion and vice-versa. Our technique combines GKS-based correlation inequalities on path space with recursive multi-scale estimates.
Paper Structure (6 sections, 16 theorems, 136 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Correlation inequalities and change of measure
  4. Proof of Theorem \ref{['thm:short_range']}
  5. Recursive variance improvement
  6. Symmetric laws under a one–variable tilt

Key Result

Theorem 1.4

Let $\hat{{\mathbb P}}_{\alpha,T}$ obey Assumption ass:short_range. Assume that there exists an even $i\in{\mathbb N}$ such that $c_{i,2}>0$. Set $\zeta := c_{i,2}$. Then, there exist constants $C>0$, $a >0$ such that for all $\alpha \ge 1$

Figures (1)

  • Figure 1: Dyadic recursion for the normalized endpoint $\frac{1}{\sqrt{T}}x_T$ when $T = 2^n$. The first levels $\sigma^{1}_{T/2},\sigma^{2}_{T/2}$ and $\sigma^{11}_{T/4},\dots,\sigma^{22}_{T/4}$ are drawn explicitly. The horizontal layer of dots indicates further dyadic levels before reaching the microscopic blocks at variance $V_1=1$. Red dashed lines indicate that we can treat variables on separate sides as independent of each other.

Theorems & Definitions (38)

  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 3.1: simon79
  • proof
  • Corollary 3.2
  • proof
  • ...and 28 more