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 $1/2$, 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).