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

TL;DR

This paper investigates a class of random homogeneous systems, demonstrating their convergence to a specific probability distribution, and applies results to series-parallel graphs and hipster random walks.

Contribution

It establishes weak convergence of a broad class of random systems to a known distribution, confirming conjectures for series-parallel graphs and analyzing hipster random walks.

Findings

Convergence to the distribution with density 3/4 (1 - x^2) on (-1,1)

Affirmative answer to conjectures on effective resistance in series-parallel graphs

Recovers previous results on hipster random walks

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 $\frac34 \, (1-x^2) \, {\bf 1}_{\{ x\in (-1, \, 1)\} }$. 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).