The critical Karp--Sipser core of random graphs

TL;DR

This paper analyzes the size of the Karp--Sipser core in random graphs with degrees 1, 2, and 3 at criticality, revealing a precise asymptotic behavior involving Brownian motion hitting times.

Contribution

It proves a conjecture about the size of the Karp--Sipser core at criticality and introduces a multi-scale analysis of the leaf-removal process near extinction.

Findings

Karp--Sipser core size scales as n^{3/5} at criticality.

Core size is proportional to the inverse square of the hitting time of a Brownian motion.

Detailed probabilistic analysis near the process's extinction time.

Abstract

We study the Karp--Sipser core of a random graph made of a configuration model with vertices of degree $1,2$ and $3$. This core is obtained by recursively removing the leaves as well as their unique neighbors in the graph. We settle a conjecture of Bauer & Golinelli and prove that at criticality, the Karp--Sipser core has size $ \approx \mathrm{Cst} \cdot \vartheta^{-2} \cdot n^{3/5}$ where $\vartheta$ is the hitting time of the curve $t \mapsto \frac{1}{t^{2}}$ by a linear Brownian motion started at $0$. Our proof relies on a detailed multi-scale analysis of the Markov chain associated to Karp-Sipser leaf-removal algorithm close to its extinction time.