The effect of disorder for pinning models is a subject which has attractedmuch attention in theoretical physics and rigorous mathematical physics. Apeculiar point of interest is the question of coincidence of the quenched andannealed critical point for a small amount of disorder. The question has beenmathematically settled in most cases in the last few years, giving inparticular 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 leftpartially open. In this paper, we give a complete answer to the question byproving a simple necessary and sufficient criterion on the return probabilityfor disorder relevance, which confirms earlier predictions from the literature.Moreover, we also provide sharp asymptotics on the critical point shift: in thecase of the pinning (or wetting) of a one dimensional simple random walk, theshift of the critical point satisfies the following high temperatureasymptotics $$ \lim_{\beta\rightarrow 0}\beta^2\log h_c(\beta)= - \frac{\pi}{2}. $$ Thisgives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus(Journal of Statistical Physics, 1992).
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