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

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 $K(n)=n^{-3/2}\varphi (n)$ where $\varphi$ 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 $$\underset{\beta \to 0}{\lim }{\beta }^2\log h_c(\beta )=-\frac{\pi }{2}.$$ This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992).

34 Pages

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• The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency
• Results and conjectures on a toy model of depinning
• Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case
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• Weak coupling limits for directed polymers in tube environments
• Free Energy of the Cauchy Directed Polymer Model at High Temperature
• Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation
• The zeros of the partition function of the pinning model