Extremes of local times for simple random walks on symmetric trees

2016/03/30 by Abe, Yoshihiro
#60G70 #60J10 #60J55 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1603.09047

Abstract

We consider local times of the simple random walk on the b-ary tree of depth n and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth r_n rooted at the (n-r_n) level, where (r_n)_{n ≥ 1} satisfies lim_{n → ∞} r_n = ∞ and lim_{n → ∞} r_n/n ∈ [0, 1). We show that the point process weakly converges to a Cox process with intensity measure αZ_∞ (dx) ⊗ e^{-2√(log b)~y}dy, where α> 0 is a constant and Z_∞ is a random measure on [0, 1] which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.

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