Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case

2015/12/16 by Lacoin, Hubert
#60K35 #60K37 #82B27 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1512.05240

Abstract

This paper continues a study initiated in [34], on the localization transition of a lattice free field on \mathbb Z^d interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential h goes past a threshold h_c: from a delocalized phase hh_c where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case (d=2) for which we had obtained so far only limited results. We prove that the value of h_c(β) is the same as for the annealed model, for all values of β and that in a neighborhood of h_c. Moreover we prove that in contrast with the case d≥ 3 where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order lim_{u→ 0+} ( log F(β,h_c(β)+u))/((log u))= ∞.

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