### 18/06/2019, 13:30 — 14:30 — Room P4.35, Mathematics Building

Pietro Caputo, *Università Roma Tre*

### The spectral gap of the interchange process: a review

Aldous’ spectral gap conjecture asserted that on any graph the random walk process and the interchange process have the same spectral gap. In this talk I will review the work in collaboration with T. M. Liggett and T. Richthammer from 2009, in which we proved the conjecture by means of a recursive strategy. The main idea, inspired by electric network reduction, was to reduce the problem to the proof of a new comparison inequality between certain weighted graphs, which we referred to as the *octopus inequality*. The proof of the latter inequality is based on suitable closed decompositions of the associated matrices indexed by permutations. I will first survey the problem, with background and consequences of the result, and then discuss the recursive approach based on network reduction together with some sketch of the proof. I will also present a more general, yet unproven conjecture.