04 April 2018

Next Mark Kac seminar

Utrecht, Janskerkhof 15a, room 001

Speaker: Anton Thalmaier (Luxembourg)
Title  : Brownian motion, Ricci curvature, functional inequalities and
geometric flows II: Characterizing Ricci curvature by
functional inequalities

Speaker: Pietro Caputo (Rome)
Title  : Non reversible random walks on sparse random structures:
stationary distribution and mixing time

Abstract Anton Thalmaier
We present recent work on characterizing Ricci curvature and Ricci flow
in terms of functional inequalities for heat semigroups. The talk
includes extensions of these methods to geometric flows on manifolds, as
well as to the path space of Riemannian manifolds evolving under a
geometric flow.

(The slides of the first lecture are available on the Mark Kac website)

Abstract Pietro Caputo
Analyzing the stationary distribution and the convergence to
stationarity of non reversible Markov chains is often a very challenging
task. In these  lectures we discuss these problems for a class of random
walks in random directed graphs and other sparse random structures. We
first consider the configuration model with prescribed in- and out-
degree sequences. The mixing time is identified explicitly in terms of
the degree sequences, while the stationary distribution is shown to have
a nontrivial shape, characterized via recursive distributional
equations. Moreover, the chain has a sharp cutoff behavior around the
mixing time, with a universal Gaussian shape inside the cutoff window.
We then discuss the extension of these results to a larger family of
sparse Markov chains whose transition matrices have exchangeable random
rows satisfying a sparsity assumption. We show that the cutoff
phenomenon holds in this general setting, with a mixing time
characterized in terms of the average one step entropy of the Markov
chain. Examples include various models of random directed graphs and
random walks among heavy tailed directed conductances.

The Mark Kac seminar is an activity of the STAR stochastics cluster. Seealso our website http://www.win.tue.nl/markkac