Upper tail large deviations of regular subgraph counts in Erd\H{o}s-R\'{e}nyi graphs in the full localized regime

TL;DR

This paper proves the conjecture that the upper tail large deviations for counts of regular subgraphs in Erdős-Rényi graphs occur due to localized structures, across the entire localized regime.

Contribution

It extends the proof of the upper tail large deviation conjecture from cliques to all connected regular graphs in the full localized regime.

Findings

Confirmed the conjectured speed of large deviations as n^2 p^Δ log(1/p).

Established the rate function via a mean-field variational problem.

Unified the understanding of large deviations for all connected regular subgraphs.

Abstract

For a $\Delta$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erd\H{o}s-R\'{e}nyi graph on $n$ vertices with edge probability $p$, has generated significant interests. For $p\ll 1$ and $np^{\Delta/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$ exceeds its expectation by a constant factor is predicted to hold at a speed $n^2 p^{\Delta} \log (1/p)$ and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of $p$, the upper tail large deviations for cliques of fixed size was proved by Harel, Mousset, and Samotij \cite{hms} in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime.