Parking on transitive unimodular graphs

TL;DR

This paper studies a parking process on transitive unimodular graphs where cars randomly walk and park, revealing a phase transition at p=1/2 for the recurrence of the root site.

Contribution

It establishes a sharp phase transition in parking behavior on transitive unimodular graphs based on the initial car density p.

Findings

Root is visited infinitely often for p ≥ 1/2

Root is visited finitely often for p < 1/2

Phase transition at p=1/2 in parking dynamics

Abstract

Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p \geq 1/2$, and only finitely many times otherwise.