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Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed

Yucheng Liu

TL;DR

The paper proves that the continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has an asymptotically deterministic escape speed $\theta(g)$, and that $\theta(g)$ is strictly increasing in the repelling strength $g$. It develops a transfer-matrix framework through a supersymmetric BFS-Dynkin isomorphism to express two-point functions and uses Tauberian theory to translate Laplace-transform asymptotics into time asymptotics, establishing both the existence of $\theta(g)$ and a precise variational formula $\theta(g)=\big(-\partial_\nu \|Q(g,\nu)\|\big|_{\nu=\nu_c(g)}\big)^{-1}$. Monotonicity of the speed is proved via stochastic dominance, showing $\theta'(g)>0$ for all $g>0$. The analysis also yields critical exponents for the two-point function, susceptibility, and correlation lengths that coincide with those of the strictly self-avoiding walk, and provides finite-volume transfer-matrix representations that underpin the infinite-volume asymptotics. Collectively, the results extend known one-dimensional results to a continuous-time setting, connect the speed to the top eigenvalue of a transfer operator, and illuminate the critical behavior of this polymer-like model.

Abstract

Weakly self-avoiding walk (WSAW) is a model of simple random walk paths that penalizes self-intersections. On $\mathbb{Z}$, Greven and den Hollander proved in 1993 that the discrete-time weakly self-avoiding walk has an asymptotically deterministic escape speed, and they conjectured that this speed should be strictly increasing in the repelling strength parameter. We study a continuous-time version of the model, give a different existence proof for the speed, and prove the speed to be strictly increasing. The proof uses a transfer matrix method implemented via a supersymmetric version of the BFS--Dynkin isomorphism theorem, spectral theory, Tauberian theory, and stochastic dominance.

Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed

TL;DR

The paper proves that the continuous-time weakly self-avoiding walk on has an asymptotically deterministic escape speed , and that is strictly increasing in the repelling strength . It develops a transfer-matrix framework through a supersymmetric BFS-Dynkin isomorphism to express two-point functions and uses Tauberian theory to translate Laplace-transform asymptotics into time asymptotics, establishing both the existence of and a precise variational formula . Monotonicity of the speed is proved via stochastic dominance, showing for all . The analysis also yields critical exponents for the two-point function, susceptibility, and correlation lengths that coincide with those of the strictly self-avoiding walk, and provides finite-volume transfer-matrix representations that underpin the infinite-volume asymptotics. Collectively, the results extend known one-dimensional results to a continuous-time setting, connect the speed to the top eigenvalue of a transfer operator, and illuminate the critical behavior of this polymer-like model.

Abstract

Weakly self-avoiding walk (WSAW) is a model of simple random walk paths that penalizes self-intersections. On , Greven and den Hollander proved in 1993 that the discrete-time weakly self-avoiding walk has an asymptotically deterministic escape speed, and they conjectured that this speed should be strictly increasing in the repelling strength parameter. We study a continuous-time version of the model, give a different existence proof for the speed, and prove the speed to be strictly increasing. The proof uses a transfer matrix method implemented via a supersymmetric version of the BFS--Dynkin isomorphism theorem, spectral theory, Tauberian theory, and stochastic dominance.
Paper Structure (5 sections, 7 theorems, 22 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The continuous-time WSAW model and main results
  3. Two-point function, susceptibility and correlation length
  4. Finite-volume two-point function
  5. Critical parameter $\nu_c$ and properties of $Q$

Key Result

Theorem 1.1

There exists an analytic function $\theta: (0, \infty) \to (0, \infty)$ with $\theta' > 0$ everywhere such that for all $g>0$, $\varepsilon>0$,

Figures (1)

  • Figure 1: Numerical evaluation of the escape speed $\theta(g)$ of the WSAW model ($\phi(t) = t^2$). The dots are the numerical results, interpolated by cubic spline. The computation is based on truncating and discretizing the integral operator $Q$ (defined in \ref{['def:Q']}) at $s=100$ and with step size $0.001$. The numerical results suggest $\theta(g) \sim Cg^{1/3}$ as $g\to 0$. For the discrete-time model, this asymptotic relation is proved by van der Hofstad and den Hollander in HH1995.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 1 more