The critical Karp--Sipser core of random graphs

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 $\approx \mathrm{C}\mathrm{s}\mathrm{t}\cdot {\vartheta }^{-2}\cdot n^{3\mathrm{/}5}$ 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.