An afternoon on random walks on groups.
14.00 – 14.45 Giulio Tiozzo (Toronto) « Roots of Alexander polynomials of random positive braids »;
15.00 – 15.45 Amaury Freslon (Orsay) « How to (badly) shuffle cards? »;
16.15 – 17.00 Charles Bordenave (Marseille) « Strong convergence of matrix algebras and applications to random walks ».
Giulio Tiozzo « Roots of Alexander polynomials of random positive braids »
As originally observed experimentally by Dehornoy, roots of Alexander polynomials of random knots display interesting patterns. In this work, joint with N. Dunfield, we prove several results on the distribution of such roots in the complex plane, and discuss further conjectures that originate from them.
Using the Burau representation, this corresponds to studying random walks on the group SL(2, C[t]) of 2-by-2 matrices with polynomial coefficients. We compute a sharp lower bound on the probability that such roots lie on the unit circle, and prove a related central limit theorem. We also show there is a large root-free region near the origin.We introduce the notion of a Lyapunov exponent for the Burau representation, in the spirit of Deroin-Dujardin, and a corresponding bifurcation measure, which we prove to be the limiting measure for the distribution of roots on a region of parameter space.
Amaury Freslon « How to (badly) shuffle cards? »
Card shuffling can be modelled by random walks on permutation groups, and the first example which was studied in depth is the one given by random transpositions. In that case, Diaconis and Shahshahani proved that the corresponding Markov chain exhibits a so-called cut-off phenomenon. Moreover, Teyssier recently computed the corresponding cut-off profile, which is remarkably simple. I will explain how one can similarly define a random walk on the « quantum permutation groups », a Hopf algebra which somehow contains the usual permutation groups. I will then report on a joint work with Teyssier and Wang where we prove the cut-off phenomenon for that process and compute the cut-off profile.
Charles Bordenave « Strong convergence of matrix algebras and applications to random walks »
We will present results on the convergence of the operator norm of random matrices of large dimension. Our random matrices are build by taking tensor products of deterministic matrices and independent Haar distributed unitary matrices or independent random permutation matrices. This class of random matrices allows for example to consider random Schreier graphs of the modular group or of Cartesian products of free groups. We will explain how these convergence results can be used to prove sharp mixing time estimates on random walks. The talk will be notably based on joint works with Benoit Collins and Hubert Lacoin.
Organized by Andrei Alpeev, Laurent Bartholdi, Anna Erschler and Panagiotis Tselekidis.
Partially supported by ERC Advanced Grant 101097307 (P.I.:Laurent Bartholdi).