Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift

TL;DR

This paper characterizes when disorder affects the critical point in a marginal pinning model, providing a precise criterion and sharp asymptotics confirming earlier physics predictions.

Contribution

It establishes a necessary and sufficient condition for disorder relevance in the marginal case and derives exact asymptotics for the critical point shift.

Findings

Disorder relevance criterion confirmed for marginal case

Sharp asymptotics for critical point shift derived

Rigorous validation of earlier physics predictions

Abstract

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, i.e. where the inter-arrival law of the renewal process is given by $K(n)=n^{-3/2}\phi(n)$ where $\phi$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning (or wetting) of a one dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$ \lim_{\beta\rightarrow 0}\beta^2\log h_c(\beta)= - \frac{\pi}{2}. $$ This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992).