Â鶹Éçmadou

Date: Thursday 17th April 2025

Abstract

We consider a product of random matrices M_n = m_1 ... m_n, where the m_k's are independent and identically distributed.

Under simple algebraic assumptions, but without moment condition, we prove exponential contraction results on the projective space similar to the ones obtained by Guivarc'h, Lepage and Raugi in the 1980's under an exponential moment assumption.

The proof relies on an effective method to describe random walks on spaces containing contracting elements rather than ergodic theory, allowing for some results to work for non-invertible matrices.

The method is heavily inspired by the pivoting technique developed by Boulanger, Mathieu, Sisto, Sert and later Gouëzel around 2020 for random walks in Gromov hyperbolic spaces.

This work also allows us to derive unexpected probabilistic results like a proof to the longstanding conjecture of almost sure convergence of the coefficients under first moment assumption.

After a quick summary of the results, we will see a description of the contraction property for products of matrices in an axiomatic way before describing the pivoting technique on the free group.