The diameter of random Schreier graphs

TL;DR

This paper proves that for a large class of random Schreier graphs derived from finite groups, the diameter is logarithmic in the size of the set, with high probability, and the bounds are nearly optimal.

Contribution

It provides a combinatorial proof establishing near-optimal bounds on the diameter of random Schreier graphs for sufficiently large generating sets.

Findings

Diameter of random Schreier graphs is O(log_k n) with high probability.

The result holds for k ≥ (log n)^{1+ε} and is essentially optimal.

The proof is combinatorial, offering a new perspective on the structure of these graphs.

Abstract

We give a combinatorial proof of the following theorem. Let $G$ be any finite group acting transitively on a set of cardinality $n$. If $S \subseteq G$ is a random set of size $k$, with $k \geq (\log n)^{1+\varepsilon}$ for some $\varepsilon >0$, then the diameter of the corresponding Schreier graph is $O(\log_k n)$ with high probability. Except for the implicit constant, this result is the best possible.