Log-concave polynomials in theory and applications (tutorial)

Abstract

Log-concave polynomials give rise to discrete probability distributions with several nice properties. In particular, log-concavity of the generating polynomial guarantees the existence of efficient algorithms for approximately sampling from a distribution and finding the size of its support. This class of distributions contains several important examples, including uniform measures over bases or independent sets of matroids, determinantal point processes and strongly Rayleigh measures, measures defined by mixed volumes in Mikowski sums, the random cluster model in certain regimes, and more. In this tutorial, we will introduce the theory and applications of log-concave polynomials and survey some of the recent developments in this area.