Hipster random walks, random series-parallel graph and random homogeneous systems

2025/11/21 by Xinxing Chen, Chen, Xinxing, Thomas Duquesne +3
Mathematics · Physics and Astronomy · #05C80 #60J80 #82B20 #82B27 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2511.16880

openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28

Abstract

We study a class of random homogeneous systems. Our main result says that under suitable general assumptions, these systems converge weakly, upon a suitable normalization, to the probability distribution with density \frac34 (1-x²) \bf 1_\ x∈ (-1, 1)\ . Two special cases are of particular interest: for the effective resistance of the critical random series-parallel graph, our result gives an affirmative answer to a conjecture of Hambly and Jordan (Adv. Appl. Probab. 2004) and further conjectures of Addario-Berry et al. (Probab.Theory Related Fields 2020) and Derrida whereas for the hipster random walk, we recover a previous result of Addario-Berry et al.~(Probab. Theory Related Fields 2020).

Citations

• Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph
• Symmetric cooperative motion in one dimension
• Hipster random walks
• Weak-disorder limit at criticality for directed polymers on hierarchical graphs
• Pemantle's min-plus binary tree
• Central limit theorems for nonlinear hierarchical sequences of random variables

Related

• Hipster random walks
• Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees
• Critical random graphs: limiting constructions and distributional properties
• Limit theorems for the empirical distribution of supercritical branching random walks on transitive graphs
• On random walk on growing graphs