arXiv subjects
Publications and source records attributed to Francesca Pieroni.
Determinantal processes provide mathematical modeling of repulsion among points. In quantum mechanics, Slater determinant states generate such processes, reflecting Fermionic behavior. This note exploits the connections between the former and the latter structures by establishing quantitative bounds in terms of trace/total variation and Wasserstein distances.
We consider the kinetic transport equation that arise in the Boltzmann-Grad limit of the two-dimensional periodic Lorentz Gas. This equation has been obtained by extending the phase space of positions and velocities through the introduction of two new variables, representing respectively the time to the next collision and the corresponding impact parameter. Here we mostly focus on the case of periodic boundary conditions on the positions space: we prove that, under suitable hypothesis, the time evolution of a probability density on the extended phase space converges to the equilibrium state with respect to the $L^p$ norm ($^*$-weakly if $p=\infty$), if such initial density is $L^p$. If $p=2$, or if the initial datum does not depend on the position, we also get more precise estimates about the rate of the approach to the equilibrium. Our proof is based on the analysis of the long time behavior of the Fourier coefficients of the solution.
We obtain new bounds for the optimal matching cost for empirical measures with unbounded support. For a large class of radially symmetric and rapidly decaying probability laws, we prove for the first time the asymptotic rate of convergence for the whole range of power exponents $p$ and dimensions $d$. Moreover we identify the exact prefactor when $p\le d$. We cover in particular the Gaussian case, going far beyond the currently known bounds. Our proof technique is based on approximate sub- and super-additivity bounds along a geometric decomposition adapted to some features the density, such as its radial symmetry and its decay at infinity.
We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant. We explicitly determine the constant, showing that the average cost is proportional to (log N)^2, where N is the number of points. Our approach relies on a geometric decomposition allowing an explicit computation of the constant. Our results illustrate the potential for exact solutions of random matching problems for many distributions defined on unbounded domains on the plane.