The diameter of random Schreier graphs
Daniele Dona, Luca Sabatini
TL;DR
The paper tackles the problem of bounding the diameter of Schreier graphs arising from a transitive action of a finite group $G$ on a set $\Omega$ of size $n$, when the generating set is chosen randomly with size $k$. It develops a purely combinatorial growth framework—centered on sphere growth and random conjugates—to show that for $(\log n)^{1+\varepsilon} \le k \le n$, the diameter satisfies $\mathrm{diam}(\mathrm{Sch}(G\circlearrowleft\Omega,S)) \le C \frac{\log n}{\log k}$ with high probability, where $C=O(\varepsilon^{-3})$ (and refinements yield $O_{\delta}(\varepsilon^{-1-\delta})$). This generalizes Roichman’s Cayley-graph diameter bound to arbitrary Schreier graphs and demonstrates the optimality of the $O(\log_k n)$ scaling up to constants for these settings. The approach relies on iterated sphere-growth arguments, a key random-conjugate intersection lemma, and a two-stage expansion mechanism to convert partial growth into a global diameter bound. The results provide a robust combinatorial toolbox for understanding growth and diameter in permutation-group actions, with implications for generation and expansion phenomena in transitive groups.
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.