Scale-free percolation mixing time

TL;DR

This paper investigates the mixing time of simple random walks on scale-free percolation graphs, revealing how hubs influence mixing speed across different phases characterized by parameters  and .

Contribution

It provides a detailed phase diagram for mixing times in scale-free percolation and introduces a bootstrap method to analyze unbounded degree scenarios.

Findings

Hubs can significantly accelerate mixing times.

The phase diagram delineates regimes with different mixing behaviors.

A novel bootstrap technique reduces complex models to simpler bounded-degree cases.

Abstract

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index $\tau-1>0$. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent $\alpha>0$ in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in $\alpha$ and $\tau$. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.