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PINNING ON A DEFECT LINE: CHARACTERIZATION OF MARGINAL DISORDER RELEVANCE AND SHARP ASYMPTOTICS FOR THE CRITICAL POINT SHIFT

  • Quentin Berger (a1) and Hubert Lacoin (a2)
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 $1/2$ , that is, where the interarrival law of the renewal process is given by $\text{K}(n)=n^{-3/2}{\it\phi}(n)$ where ${\it\phi}$ 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 of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$\begin{eqnarray}\lim _{{\it\beta}\rightarrow 0}{\it\beta}^{2}\log h_{c}({\it\beta})=-\frac{{\it\pi}}{2}.\end{eqnarray}$$ This gives a rigorous proof to a claim of Derrida, Hakim and Vannimenus (J. Stat. Phys. 66 (1992), 1189–1213).

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