Marginal relevance for the $\gamma$-stable pinning model

TL;DR

This paper examines how disorder affects the critical point in a $eta$-stable pinning model with renewal jumps, especially at the boundary case where the tail exponents are critical, providing new insights into disorder relevance.

Contribution

It extends previous work by analyzing the boundary case $eta=1-rac{1}{ heta}$, showing that disorder shifts the critical point and estimating its magnitude.

Findings

Disorder shifts the critical point at the boundary case.

The magnitude of the shift is explicitly estimated.

The results clarify the role of tail exponents in disorder relevance.

Abstract

We investigate disorder relevance for the pinning of a renewal when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. Assuming that the renewal jumps have power-law decay, we determine under which condition the critical point of the system modified by the introduction of a small quantity of disorder. In an earlier study of the problem, we have shown that the answer depends on the value of the tail exponent $\alpha$ associated to the distribution of renewal jumps: when $\alpha>1-\gamma^{-1}$ a small amount of disorder shifts the critical point whereas it does not when $\alpha<1-\gamma^{-1}$. The present paper is focused on the boundary case $\alpha=1-\gamma^{-1}$. We show that a critical point shifts occurs in this case, and obtain an estimate for its intensity.