Plenary session 4

Thursday, 7/17, 9:00-12:00, Congress Centre

On the derivation of mean-curvature flow and its fluctuations from microscopic interactions

Sunder Sethuraman (University of Arizona)

The emergence of mean-curvature flow of an interface between different phases or populations is a phenomenon of long standing interest in statistical physics. In this talk, we review recent progress with respect to a class of reaction-diffusion stochastic particle systems on an \(n\)-dimensional lattice. In such a process, particles can move across sites as well as be created/annihilated according to diffusion and reaction rates. These rates will be chosen so that there are two preferred particle mass density levels \(a_1\), \(a_2\).

In the evolution, one may understand, when the diffusion and reaction schemes are appropriately scaled, that a rough interface forms between the regions where the mass density is close to \(a_1\) or \(a_2\). Via notions in the theory of hydrodynamic limits, we discuss when the scaled limit of the particle mass density field in \(n\geq 2\) is a sharp interface flow by mean-curvature. We also discuss the fluctuation field limit of the mass near the forming interface, which informs on the approach to the continuum view in a certain stationary regime in \(n=1,2\).

Minimal surfaces in a random environment

Ron Peled (University of Maryland and Tel Aviv University)

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems, first-passage percolation models and minimal cuts in the Z^D lattice with random capacities. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an “independent" or "Brownian” random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations. In particular, we prove, for all values of n, that the surfaces are delocalized in dimensions d ≤ 4 and localized in dimensions d ≥ 5. Moreover, the surface delocalizes with power-law fluctuations when d ≤ 3 and with sub-power-law fluctuations when d = 4. Many of our results are new even for d = 1 (indeed, even for d = n = 1), corresponding to the well-studied case of (non-integrable) first-passage percolation. Based on joint works with Barbara Dembin, Dor Elboim and Daniel Hadas, with Michal Bassan and Shoni Gilboa and with Michal Bassan and Paul Dario.

Bootstrap percolation and kinetically constrained models: universality results

Cristina Toninelli (Ceremade, University Dauphine - PSL and CNRS)

Recent years have witnessed significant progress in the study of bootstrap percolation (BP) models. In the initial configuration sites are occupied with probability p. The evolution of BP proceeds in discrete time: empty sites remain empty, while occupied sites become empty if and only if a certain model-dependent neighborhood is already empty. On Z^d there is now a fairly complete understanding of the dynamics starting from random initial conditions, along with a clear universality picture for their critical behavior. Much less is known about their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In these models each vertex is either infected or healthy and, iff it is infectable according to the BP rules, its state is resampled (independently) at rate one and becomes infected with probability p, and healthy with probability 1-p. These models, introduced and intensively studied in physics literature as toy models of the liquid/glass transition, present both challenging and fascinating mathematical problems. Indeed, the presence of constraints induce non-attractiveness, multiple invariant measures, and the breakdown of many powerful tools (such as coercive inequalities, coupling arguments, and censoring techniques) typically used to study convergence to equilibrium. In this talk, I will present a series of results that establish the full universality picture of KCM in two dimensions. We will see that, compared to those of BP, the universality classes for the stochastic dynamics are richer and the critical time scales diverge more rapidly due to the dominant role of energy barriers. The seminar is based on joint works with I.Hartarsky, L.Marêché, F.Martinelli, and R.Morris.