Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

2022/12/25 by Shuxiong Zhang, Zhang, Shuxiong
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2212.12835

Abstract

Let \Z_n\_{n≥ 0} be a d-dimensional supercritical branching random walk started from the origin. Write Z_n(S) for the number of particles located in a set S⊂ℝ^d at time n. Denote by R_n:=inf\ρ:Z_i(\|x|≥ ρ\)=0,∀~0≤ i≤ n\ the range of \Z_n\_{n≥ 0} before time n. In this work, we show that under some mild conditions R_n/n converges in probability to some positive constant x^* as n→∞. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates of ℙ(R_n≤ xn)~for~x∈(0,x^*);~ℙ(R_n≥ xn) ~for~ x∈(x^*,∞) as n→∞. As a by-product, we confirm a conjecture of Engländer \citeEnglander04.

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