Surface Order Large Deviations Research Articles

Phase Transitions for phi ^4_3

We establish a surface order large deviation estimate for the magnetisation of low temperature phi ^4_3. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the phi ^4_3 singular stochastic PDE. Our main technical contributions are contour bounds for phi ^4_3, which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of phi ^4_3 whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for phi ^4_3 developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020).

A coarse graining for the Fortuin–Kasteleyn measure in random media

By means of a multi-scale analysis we describe the typical geometrical structure of the clusters under the FK measure in random media. Our result holds in any dimension d ⩾ 2 provided that slab percolation occurs under the averaged measure, which should be the case for the whole supercritical phase. This work extends that of Pisztora [A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (4) (1996) 427–466] and provides an essential tool for the analysis of the supercritical regime in disordered FK models and in the corresponding disordered Ising and Potts models.

Shannon–McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations

The notion of a surface-order specific entropy h c (P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon–McMillan theorem. We obtain a representation of h c (P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Follmer and Ort’s lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behaviour in the phase-transition regime.

Surface order large deviations for 2D FK-percolation and Potts models

By adapting the renormalization techniques of Pisztora (Probab. Theory Relat. Fields 104 (1996) 427), we establish surface order large deviations estimates for FK-percolation on Z 2 with parameter q⩾1 and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.

Surface order large deviations of phase interfaces for the continuum Widom-Rowlinson model in high density limit

We establish a surface order large deviation principle characterising, in the phase coexistence region, the exponential decay rates for the probabilities of macroscopic fluctuations of phase-separating interfaces for the continuum Widom-Rowlinson binary gas, with the thermodynamic and high fugacity limits taken simultaneously. The large deviation rate function is given by an isotropic surface energy functional and hence it attains its minimum for balls which are the most favourable shapes of ‘droplets’ of dominated phase within the ‘ocean’ of dominating phase.

On the Wulff crystal in the Ising model

We study the phase separation phenomenon in the Ising model in dimensions $d \geq 3$. To this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization $m^*$. We confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase. Moreover, the rescaled droplet is asymptotically close to a definite deterministic shape, the Wulff crystal, which minimizes the surface free energy. In the course of the proof we establish a surface order large deviation principle for the magnetization. Our results are valid for temperatures $T$ below a limit of slab-thresholds $\hat{T}_c$ conjectured to agree with the critical point $T_c$. Moreover, $T$ should be such that there exist only two extremal translation invariant Gibbs states at that temperature, a property which can fail for at most countably many values and which is conjectured to be true for every $T$. The proofs are based on the Fortuin–Kasteleyn representation of the Ising model along with coarse-graining techniques.To handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis. Finally,we propose a heuristic picture that for subcritical temperatures close enough to $T_c$, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.

Surface order large deviations for Ising, Potts and percolation models

We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford≧3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq≧1 andd≧3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.

Surface order large deviations for high-density percolation

We derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a Bernoulli percolation process restricted to a big finite box, with sufficiently large parameter. We also establish a useful version of the isoperimetric inequality, which is the main tool of our proofs.

Surface order large deviations for high-density percolation

Surface order large deviations for high-density percolation

Surface order large deviations for Ising, Potts and percolation models

Surface order large deviations for Ising, Potts and percolation models

Large deviations for discrete and continuous percolation

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Large deviations for discrete and continuous percolation

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Critical Large Deviations for Gaussian Fields in the Phase Transition Regime, I

We investigate large deviations for the empirical distribution functional of a Gaussian random field on $\mathbb{R}^{\mathbb{Z}^d}, d \geq 3$, in the phase transition regime. We first prove that the specific entropy governs an $N^d$ volume order large deviation principle outside the Gibbsian class. Within the Gibbsian class we derive an $N^{d-2}$ capacity order large deviation principle with exact rate function, and we apply this result to the asymptotics of microcanonical ensembles. We also give a spins' profile description of the field and show that smooth profiles obey $N^{d-2}$ order large deviations, whereas discontinuous profiles obey $N^{d-1}$ surface order large deviations.