Law of Iterated Logarithm for random graphs

TL;DR

This paper extends the law of the iterated logarithm (LIL) to various functionals of random graphs and hypergraphs, including subgraph counts, perfect matchings, and Hamilton cycles, with new large deviation bounds.

Contribution

It proves LIL for subgraph counts, perfect matchings, and Hamilton cycles in random graphs and hypergraphs, introducing new large deviation bounds for these functionals.

Findings

LIL holds for the number of copies of a fixed subgraph H.

LIL and CLT are established for the number of Hamilton cycles in hypergraphs.

New large deviation bounds are developed for key graph functionals.

Abstract

A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$ $$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{\sigma_n \sqrt {2 \log \log n }} =1 \right] =1 . $$ In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph $H$. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random $k$-uniform hypergraphs, we obtain the Central Limit Theorem (CLT) and LIL for the number of Hamilton cycles.