Crystallization in the Winterbottom Shape and Sharp Fluctuation Laws

Abstract

Abstract We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength $$\beta >0$$ β > 0 couples the particles and the substrate. Our first main result proves crystallization for all $$\beta >0$$ β > 0 , corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) . We obtain bounds on the distance between distinct minimizers with the same number N of particles, showing a sharp scaling law $$N^{3/4}$$ N 3 / 4 when $$\beta $$ β is rational, and $$N^{1/3}$$ N 1 / 3 when $$\beta $$ β is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit.