Speaker: Riccardo Cristoferi (Radboud University)
Title: Crystallization into the triangular lattice in the plane for a family of concave entropies
Many optimal location problems in economics, signal compression, and numerical integration require to approximate the Lebesgue measure in a set with a discrete measure. The error of such an approximation can be quantified by using the p Wasserstein distance. Existence can be guaranteed if an explicit or implicit (by using the so called entropies) bound on the number of particles is imposed, say with a parameter L. For a fixed value of L the geometry of the set has a strong influence and it is therefore difficult to characterize minimizing configurations. As L increases, though, optimal configurations are expected to arrange, at least locally, into regular periodic patterns. This phenomenon is known as crystallization. It is important to determine such patterns as well as the rate of convergence of the energy to the minimum.
In this talk we consider a system of particles in the plane interacting via the 2-Wasserstein distance and where the entropy belongs to a class of concave functions. We prove that the triangular lattice gives, asymptotically, the optimal energy and we establish the asymptotic quantization error. This extends a result by Bourne, Peletier and Theil (2014) and proves a case of a conjecture by Bouchitté, Jimenez and Mahadevan (2011).
This talk is based on a joint work with David Bourne (Heriot-Watt University).