Formation of Facets for an Effective Model of Crystal Growth

Abstract

We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a 2+1 solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to "grow" through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.

Tightness and Line Ensembles for Brownian Polymers Under Geometric Area Tilts

“…[16,15] for an effective finite scale description of level lines in terms of random walks. Growing number of macroscopic level lines and multi-body interactions between them pose additional challenges [15,14].…”

Confinement of Brownian polymers under geometric area tilts

“…The loops L h can be identified via constrained Wulff variational principles: each single L h is a rescaled Wulff plaquette such as that already studied in the context of 2D Ising model [21]. A related low temperature SOS-type model which, under appropriate rescaling, features a stack of identical Wulff plaquettes was studied in [12]. In our case nested loops L h are strictly ordered by inclusion.…”

The Airy$_2$ process and the 3D Ising model

“…The caricature of the process of the crystal growth was considered in [17]. The motivation was to study the dynamics of the large droplet of the (+)phase floating in the cubic box V N of size 8N 3 filled with the (−)-phase of the low-temperature 3D Ising model, when the volume of the droplet (being of the order of cN 3 ) grows.…”

“…and the problem was to study the properties of Γ as a function of the value of C. A technical simplification was made in [17], by passing to the SOSapproximation of the model, so the level of the study corresponds to the random walk model in Section 1, rather than to Section 2.1.…”

“…The main result of [17] is that the interesting behavior of the surface Γ happens if one varies the constant C on the scale N 2 , i.e. considers the dependence of Γ as a function of the parameter a, where C = aN 2 .…”