  The effect of disorder on pinning and wetting models has attracted much
attention in theoretical physics. In particular, it has been predicted on the
basis of the Harris criterion that disorder is relevant (annealed and quenched
model have different critical points and critical exponents) if the return
probability exponent alpha, a positive number that characterizes the model, is
larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e.
coinciding critical points and exponents) if alpha < 1/2. Recent mathematical
work has put these predictions on firm grounds. In renormalization group terms,
the case alpha = 1/2 is a 'marginal case' and there is no agreement in the
literature as to whether one should expect disorder relevance or irrelevance at
marginality. The question is particularly intriguing also because the case
alpha = 1/2 includes the classical models of two-dimensional wetting of a rough
substrate, of pinning of directed polymers on a defect line in dimension (3+1)
or (1+1) and of pinning of an heteropolymer by a point potential in
three-dimensional space. Here we prove disorder relevance both for the general
alpha = 1/2 pinning model and for the hierarchical version of the model
proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), in the sense
that we prove a shift of the quenched critical point with respect to the
annealed one. In both cases we work with Gaussian disorder and we show that the
shift is at least of order exp(-1/\beta^4) for beta small, if beta is the
standard deviation of the disorder.
