Let Gd be the semi-direct product of R*+ and Rd, d ≥ 1 and let us consider the product group Gd,N = Gd x RN, N ≥ 1. For a large class of probability measures μ on Gd,N, one prove that there exists ρ(μ) ∈ ]0,1] such that the sequence of finite measures{(n(N+3)/2 / ρ(μ)n) μ*n}n ≥ 1converges weakly to a non-degenerate measure.
We consider some random band matrices with band-width whose entries are independent random variables with distribution tail in . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when , the largest eigenvalues have order , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov...
Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities of return to any neighborhood V of the neutral element decaying at infinity almost as fast as the exponential function n ↦ exp(-n). We also show that for some discrete groups , the decay of the function can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .