2022/06/10 by Ion Grama, Grama, Ion, Sebastian Mentemeier +3
Mathematics · #60J05 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Primary 60J80 #Probability (math.PR) #Secondary 60B20 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2206.04941
openalex publication_date 2022/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a branching random walk (Gu)u∈ \mathbb T on the general linear group \textrmGL(V) of a finite dimensional space V, where \mathbb T is the associated genealogical tree with nodes u. For any starting point v ∈ V ∖\0\ with ‖v‖=1 and x = \mathbb R v ∈ \mathbb P(V), let Mxn=max|u| = n log ‖ Gu v ‖ denote the maximal position of the walk log ‖ Gu v ‖ in the generation n. We first show that under suitable conditions, limn → ∞ (Mnx )/(n) = γ almost surely, where γ∈ \mathbb R is a constant. Then, in the case when γ= 0, under appropriate \textit boundary conditions, we refine the last statement by determining the rate of convergence at which Mnx converges to -∞. We prove in particular that limn → ∞ (Mnx)/(log n) = -(3)/(2α) in probability, where α>0 is a constant determined by the boundary conditions. Analogous 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.