Crystallization in the Winterbottom shape and sharp fluctuation laws

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 $\beta>0$ couples the particles and the substrate. Our first main result proves crystallization for all $\beta>0$, 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 $\beta\in (0,1)$. We obtain bounds on the distance between distinct minimizers with the same number $N$ of particles, showing a sharp scaling law $N^{3/4}$ when $\beta$ is rational, and $N^{1/3}$ when $\beta$ 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.