Upper tails for triangles

TL;DR

This paper derives tight exponential bounds for the probability that the number of triangles in an Erdős-Rényi random graph exceeds its expectation by a certain factor, highlighting the behavior of upper tail deviations.

Contribution

It provides a precise exponential inequality for the upper tail of triangle counts in Erdős-Rényi graphs, extending understanding of large deviations in random graph theory.

Findings

Established exponential decay bounds for triangle count deviations.

Proved bounds are tight up to a constant factor.

Enhanced understanding of rare event probabilities in random graphs.

Abstract

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) $$\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to the value of $C_{\eta}$.