The disordered lattice free field pinning model approaching criticality

Abstract

We continue the study, initiated in [21], of the localization transition of a lattice free field φ = (φ(x)) x∈Z d , d ≥ 3, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is 2σ 2 d | log(h − hc(β))| to leading order when h hc(β) except on a vanishing fraction of sites (σ 2 d is the single site variance of the free field).

Solid-On-Solid interfaces with disordered pinning

“…A first major difference with the homogeneous model is that the free energy growth at criticality is quadratic in h − h c instead of linear in the homogeneous case. This change of power exponent (from 1 to 2) is analogous to what was observered for the Lattice Gaussian Free Field (LGFF) in dimension larger than 3 [25] (when d ≥ 3 the variance of the Lattice Free Field is uniformly bounded which makes the model somehow similar to SOS in the rigid phase, the two dimensional LGFF displays a very different behavior, see [32] for details). Also, and this is perhaps the most novel aspect of our result, we identifiy a phenomenon which is specific to the discrete nature SOS: the asymptotic of the free energy is not a pure power.…”

“…The case which offers most similarity with low temperature SOS is that of dimension d ≥ 3 (for which it is known that the variance of the field is bounded). In that case it was shown in [23,25] that while the value of the critical point h c is not affected by the introduction of inhomogeneities, disorder smoothens the phase transition. The homogeneous model displays a phase transition of first order (like the model studied in the present paper, cf.…”

“…More recently an extensive answer to the question of disorder relevance has been given for higher dimensional surface models, in the case where φ is the lattice Gaussian Free Field (GFF) on Z d (and δ x is replaced by [15,23,24,25,32]. The case which offers most similarity with low temperature SOS is that of dimension d ≥ 3 (for which it is known that the variance of the field is bounded).…”

“…In the specific case of pinning models, the study of disorder relevance for one dimensional interfaces has given rise to a plentiful literature partly motivated by a connection the DNA denaturation phenomenon (see [21,22] for a review and references). The case of higher dimension was explored only more recently in the case of lattice free field interfaces [15,23,24,25,32] for which heights takes values in R.…”

Localization, big-jump regime and the effect disorder for a class of generalized pinning models

“…Unless we specify otherwise, when we speak of pinning models, like here, we mean one dimensional pinning models: there are several higher dimensional generalizations that can and have been considered (e.g. [32] and references therein), and a class is going to be very relevant to us and will soon be mentioned; • the model exhibits a transition between a delocalized and a localized regime which is understood in depth thanks to solvability. Notably, the model depends on a real parameter α ≥ 0 and the critical phenomenon depends on α is such a way that the (de)localization transition can be of arbitrary order, i.e.…”