2026/07/17 by Benjamin Braun, Max Hlavacek, Cesar J. Meza +2 · 1 citation
#math.CO
Every polynomial with real non-negative coefficients yields a finite probability distribution after normalization. The Ehrhart h^*-polynomial of a lattice polytope P is a non-negative integer polynomial that encodes the integer-point counts for positive integer dilations of P. We study the corresponding finite distributions, which we call h^*-distributions. We determine the mean and variance of these distributions, establish a connection between higher moments and Ehrhart polynomial coefficients, and study their cluster points in the d-dimensional probability simplex. We consider the special case of real-rooted h^*-distributions, applying existing tail bounds to obtain new linear inequalities for the coefficients of real-rooted h^*-polynomials arising from reflexive polytopes. We conclude by establishing sufficient conditions under which a sequence of real-rooted h^*-distributions is asymptotically normal, and we apply our results to various families of polytopes, including zonotopes and Pitman-Stanley polytopes.