The upper tail problem for induced 4-cycles in sparse random graphs

TL;DR

This paper determines the asymptotic behavior of the upper tail probability for induced 4-cycles in sparse random graphs, revealing a new large deviation phenomenon where naive approximations fail.

Contribution

It extends techniques from prior work on clique upper tails to induced 4-cycles, uncovering a novel deviation phenomenon in sparse graphs.

Findings

Asymptotic formulas for upper tail probabilities of induced 4-cycles.

Identification of a new large deviation phenomenon in sparse graphs.

Failure of naive mean-field approximation in certain p ranges.

Abstract

Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4-cycle in the binomial random graph $G_{n,p}$. We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain (large) range of $p$, the upper tail of the induced 4-cycle does not admit a naive mean-field approximation.