# Formation of Facets for an Effective Model of Crystal Growth

## TL;DR

This paper models microscopic facet formation in crystal growth, revealing a sequence of phase transitions at low temperatures driven by particle constraints, with detailed concentration results near transition points.

## Contribution

It introduces a new effective 2+1 solid-on-solid model coupled with Bernoulli fields to analyze facet formation and phase transitions in crystal growth.

## Key findings

- Infinite sequence of first order phase transitions at low temperatures.
- Sharp concentration of level lines around variational solutions.
- Model captures spontaneous monolayer creation during 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.