THE EXTREMAL POSITION OF A BRANCHING RANDOM WALK IN THE GENERAL LINEAR GROUP

Overview & Synopsis

Consider a branching random walk (Gu) u∈T on the general linear group GL(V) of a finite dimensional space V , where T is the associated genealogical tree with nodes u. For any starting point v ∈ V \ {0}, let M v n = max |u|=n log Guv denote the maximal position of the walk log Guv in the generation n. We first show that under suitable conditions, limn→∞ M v n n = γ+ almost surely, where γ+ ∈ R is a constant. Then, in the case when γ+ = 0, under appropriate boundary conditions, we refine the last statement by determining the rate of convergence at which M v n converges to −∞. We prove in particular that limn→∞ M v n log n = − 3/(2α) in probability, where α > 0 is a constant determined by the boundary conditions. Similar properties are established for the minimal position. As a consequence we derive the asymptotic speed of the maximal and minimal positions for the coefficients, the operator norm and the spectral radius of Gu.