The Second Phase Transition of the Contact Process on a Random Regular Graph
John Fernley
TL;DR
The paper investigates the second phase transition of the contact process on large random $d$-regular graphs, whose local limit is the $d$-regular tree. It couples the finite graph to the tree via a configuration-model framework and derives sharp concentration and moment bounds for the tree process, including $\mathbb{E}(|\xi_t|)=\Theta(e^{c_\lambda t})$ and $\mathbb{E}(|\xi_t|^n)=O(t^{n\log_2(d/(d-2))} e^{n c_\lambda t})$, along with a left-tail control for growth. The authors define reinfection times $I_k$ and show a dichotomy: for $\lambda>\lambda_2(d)$ reinfections occur at linear scale, $I_{\lfloor\varepsilon N\rfloor}=\Theta_P(\varepsilon N)$, while for $\lambda_1(d)<\lambda<\lambda_2(d)$ reinfections are delayed by a healthy period, $I_{\lfloor\varepsilon N\rfloor}=\frac{1}{c_\lambda}N+\Theta_P(\varepsilon N)$. Core innovations include a severed-pioneer framework and a detailed infection-path construction that links forward and backward infection histories, establishing the finite-graph manifestation of the tree's double phase transition. These results provide a locality-type bridge between the infinite-volume phase diagram and finite sparse networks, with potential applications to understanding reinfection dynamics in large-scale networks.
Abstract
The regular tree corresponds to the random regular graph as its local limit. For this reason the famous double phase transition of the contact process on regular tree has been seen to correspond to a phase transition on the large random regular graph, at least at the first critical value. In this article, we find a phase transition on that large finite graph at the second critical value: between linear reinfections and reinfections following a long healthy period.