Products of Beta matrices and sticky flows

In [2], a family of stochastic flows of kernels on S 1 called " sticky flows " is described. Sticky flows are defined by their " moments " which are consistent systems of transition kernels on S 1. In this note, a discrete version of sticky flows is presented in the case the sticky flows are associated with a system of Brownian particles on S 1. This discrete model is defined by products of Beta matrices on the discrete torus Z/NZ and will be called a Beta flow. Similarly to the continuous case, the moments of the Beta flow are consistent systems of transition matrices on Z/NZ. A convergence of the Beta matrices to sticky kernels is shown at the level of the moments. 1 Beta matrices and Polya scheme Let a be a positive parameter and N be an even positive integer. We define a random transition matrix K on the discrete torus T N = Z/NZ as follows : a N , a N) random variables. Let (K n) n be a i.i.d sequence of such random transition matrices and let {Z(t), t ≥ 0} be an independent Poisson process on R with intensity N 2. The family of matrices (K N,s,t) s≤t defined by: K N,s,t = K Z(s)+1 K Z(s)+2 · · · K Z(t) for every s ≤ t , is a stochastic flow of kernels on Z/NZ. It will be called the Beta flow on Z/NZ.