  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).
