Upper Tails of Subgraph Counts in Sparse Regular Graphs

TL;DR

This paper investigates the probability of observing significantly more copies of a fixed subgraph in sparse random regular graphs than expected, providing precise asymptotic behavior for most graphs and highlighting exceptions.

Contribution

It determines the upper tail probabilities for subgraph counts in sparse regular graphs within a logarithmic gap, including new behaviors for specific graphs.

Findings

Upper tail probabilities are characterized within a logarithmic gap for most graphs.

For graphs with average degree > 4, the asymptotics are determined up to a 1+o(1) factor.

An example is provided where the behavior differs from previous models in the same sparsity regime.

Abstract

What is the probability that a sparse $n$-vertex random $d$-regular graph $G_n^d$, $n^{1-c}<d=o(n)$ contains many more copies of a fixed graph $K$ than expected? We determine the behavior of this upper tail to within a logarithmic gap in the exponent. For most graphs $K$ (for instance, for any $K$ of average degree greater than $4$) we determine the upper tail up to a $1+o(1)$ factor in the exponent. However, we also provide an example of a graph, given by adding an edge to $K_{2,4}$, where the upper tail probability behaves differently from previously studied behavior in both the sparse random regular and sparse Erd\H{o}s-R\'{e}nyi models in this sparsity regime.