On the lower tail variational problem for random graphs

TL;DR

This paper investigates the probability of significantly fewer subgraphs than expected in a random graph, characterizing the dominant configurations and phase transitions in the lower tail large deviation problem.

Contribution

It provides a partial characterization of the replica symmetric phase for the lower tail problem in random graphs and identifies when Erdős-Rényi graphs dominate the lower tail behavior.

Findings

Main contribution from Erdős-Rényi graphs with tilted edge density for small p and δ below δ_H.

Breakdown of this behavior for non-bipartite graphs and δ close to 1.

Partial understanding of the replica symmetric phase in the variational problem.

Abstract

We study the lower tail large deviation problem for subgraph counts in a random graph. Let $X_H$ denote the number of copies of $H$ in an Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed $0 < \delta < 1$. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for $p \ge n^{-\alpha_H}$ (and conjecturally for a larger range of $p$). We study this variational problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every $H$, and $0 < \delta < \delta_H$ for some $\delta_H > 0$, as $p \to 0$ slowly, the main contribution to the lower tail probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite $H$ and $\delta$ close to 1.