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

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 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 of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics lim(beta -> 0)beta(2) log h(c) = -pi/2. This gives a rigorous proof to a claim of Derrida, Hakim and Vannimenus (J. Stat. Phys. 66 (1992), 1189-1213).