Pinning and disorder relevance for the lattice Gaussian free field

Abstract

Abstract. This paper provides a rigorous study of the localization transition for a Gaussian free field on 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 and one can show that a localization-delocalization transition takes place when varying the average pinning potential h: the free energy density is zero in the delocalized regime, that is for h smaller than a threshold hc, and it is positive for h > hc. For d ≥ 3 we compute hc and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by ((h−hc)+) 2 times a positive constant and that, for Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an example in which a relevant disorder critical exponent can be made explicit: in theoretical physics disorder is said to be relevant when the disorder changes the critical behavior of a system and, while there are cases in which it is known that disorder is relevant, the exact critical behavior is typically unknown. For d = 2 we are not able to decide whether the quenched and annealed critical points coincide, but we provide an upper bound for the difference between them.

Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger

“…These results offer a sharp contrast with those obtained in the absence of half-space repulsion [11,13] (see also [6] for a first contribution to the subject). In that case, the transition, which is of first order when d ≥ 3 and of second order when d = 2 for the homogeneous case, becomes smoother in the disordered one (order two and infinity respectively).These difference can be interpreted in the light of Harris criterion concerning disorder relevance [12]: for the wetting transition here, the homogeneous model has a smooth transition (the specific heat exponent is negative), and for this reason, disorder should be irrelevant, i.e.…”

“…Of course Pφ Λ is the finite volume LGFF. Much has been written about this field: we stress here that for d ≥ 3 the N → ∞ limit, with respect to the product topology, of P u Λ N exists and it can be characterized as the Gaussian field with constant expectation u and covariance of φ x and φ y equal to the expected time spent in y by a simple symmetric random walk issued from x (for more on this very well known issue we refer to [11,Sec. 2.9] and references therein).…”

“…It is worth pointing out that also in the case K = 0, treated in [11], we have that h c (β) = 0, but (2.11) does not hold! In fact in [11] it is shown that the critical behavior in that case has a power law behavior. More importantly, in [11] it is shown that in the K = 0 case disorder is relevant, i.e.…”

“…In fact in [11] it is shown that the critical behavior in that case has a power law behavior. More importantly, in [11] it is shown that in the K = 0 case disorder is relevant, i.e. it changes the critical behavior (for any β > 0) -while (2.11) shows disorder irrelevance (more on this just below).…”

“…it should not change the critical behavior, at least for small perturbation. For the pinning transition studied in [11,13] the specific heat exponent is positive, so the Harris criterion predicts disorder relevance. Note that the model is also defined in dimension one: in that case the harmonic crystal is simply a random walk with IID Gaussian increments.…”