Disorder relevance for pinning of random surfaces
Hubert Lacoin (IMPA)
(joint work with G. Giacomin)
Disorder relevance is an important question in Statistical Mechanics. It can be formulated as follows: "If the Hamiltonian of model is modified by adding a small random perturbation, does it conserve a phase transition with the same characteristics as that of the pure model." A mathematical investigation of this matter is of course possible only for models for which the phase transition is rigorously understood in the pure setup, and our work concerns a very simple and tractable model of surfaces in interaction with a defect plane.
The surfaces is modeled by the graph of a Gaussian-Free-Field $\mathbb Z^d$, $d\ge 2$, and the interaction is given by an energy reward for each point of the graph whose height is in the interval $[-1,1]$. The system undergoes a wetting transition from a localized phase to a delocalized one, when the mean energy of interaction varies.
We investigate the modification of the free-energy curve induced by the introduction of inhomogeneity in the interaction. We show that in a certain sense the critical point is left invariant by the presence of homogeneity, but that the localization transition becomes much smoother.