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A connection between the random pinning model and random walks in sparse random environments

Julien Poisat

Abstract

The purpose of this short note is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged on the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.

A connection between the random pinning model and random walks in sparse random environments

Abstract

The purpose of this short note is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged on the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.

Paper Structure

This paper contains 5 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Reminder on random walks in a potential
  3. Random walk in a random environment (RWRE)
  4. Random walk in a sparse random environment (RWsRE)
  5. Connection with the random pinning model

Key Result

Proposition 1

Assume that $\mathrm{E}(\log \tau_1) < \infty$. Then the RWsRE is transient under the annealed law, i.e. whenever $h<0$Same remark as above..

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • proof : Proof of Proposition \ref{['pr:key-identity']}
  • Proposition 3
  • Proposition 4
  • proof : Proof of Proposition \ref{['pr:dichotomy2']}
  • Remark 5
  • Theorem 6
  • proof : Proof of Theorem \ref{['thm:rwrse-thresholds']}
  • Remark 7