Trickle-Down Theorems and Local-To-Global Analysis of Markov Chains

Abstract

This thesis covers multiple results related to high dimensional counting and samplingproblems, as well as the broader theory of high dimensional expanders. Central to these re- sults are "local-to-global" phenomena that allow the study of high dimensional distributions through multiple forms of localization. In particular, we prove new trickle-down theorems and apply them to problems in different fields. This includes proving rapid mixing of the natural Markov chain for sampling from graph colorings in a previously unsolved regime, and obtaining significantly improved bounds on the local spectral expansion of recent con- structions of sparse high dimensional expanders, which are of particular interest in coding theory and complexity theory. We also use a local-to-global perspective to provide evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly. Finally, we apply a local-to-global technique to take a step towards characterizing the coef- ficients of homogeneous completely log-concave polynomials, which also implies fast mixing of the natural random walk for sampling from the high dimensional distributions associated with such polynomials.