2020/11/14 by Eleanor Archer, Archer, Eleanor
Computer Science · Mathematics · Physics and Astronomy · #60J10 #60J35 #60J80 #60K37 (primary) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2011.07266
openalex publication_date 2020/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree n with an independent copy of a graph Gn and gluing the inserted graphs along the tree structure. We assume that there exist constants d, R ≥ 1, v < ∞ such that the diameter, effective resistance across and volume of Gn respectively grow like n(1)/(d), n(1)/(R), nv as n → ∞. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like cx-α for some constant c>0 and some α∈ (1,2). We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of α, d, R and v, along with bounds on the fluctuations of these quantities.