Random walk versus random line

TL;DR

This paper analyzes the behavior of certain random walks with drift and their equivalence to Solid-On-Solid models, exploring phase transitions related to wetting phenomena and phase diagrams.

Contribution

It revisits the law equivalence between random walk bridges and Solid-On-Solid models, detailing phase diagrams and wetting transitions based on drift parameters.

Findings

Complete wetting occurs for delta <= 1.

Critical partial wetting occurs for delta > 1.

Phase diagrams are characterized by recurrence and wetting regimes.

Abstract

We consider random walks X_n in Z+, obeying a detailed balance condition, with a weak drift towards the origin when X_n tends to infinity. We reconsider the equivalence in law between a random walk bridge and a 1+1 dimensional Solid-On-Solid bridge with a corresponding Hamiltonian. Phase diagrams are discussed in terms of recurrence versus wetting. A drift -delta/X_n of the random walk yields a Solid-On-Solid potential with an attractive well at the origin and a repulsive tail delta(delta+2)/(8X_n^2) at infinity, showing complete wetting for delta<=1 and critical partial wetting for delta>1.