Abstract
This tutorial will focus on a generalization of real stable generating polynomials called complete log-concave polynomials. In this generalization, the strict requirements on the locations of roots are dropped and replaced instead by the weaker property of log-concavity (as a function) over the positive orthant.
Complete log-concavity captures a large class of distributions and their associated generating polynomials, which includes uniform measures over the bases or independent sets of matroids, volumes of Minkowski sums, the random cluster model in the ``negative dependence’' regime, and certain powers of determinantal point processes.
We will survey methods to verify complete log-concavity as well as operations that preserve it. We will see some applications of complete log-concavity, including a proof of Mason’s ultra-log-concavity conjecture on the number of independent sets in matroids.