On the scaling limit of finite vertex transitive graphs with large diameter

TL;DR

This paper investigates the geometric limits of large, finite, vertex transitive graphs with sub-polynomial growth, showing they converge to tori or circles under rescaling, using advanced group theory and elementary methods.

Contribution

It establishes new convergence results for vertex transitive graphs with sub-polynomial growth, linking graph properties to geometric limits like tori and circles.

Findings

Graphs converge to tori of dimension less than q

Roughly transitive graphs converge to a circle

Uses recent quantitative Gromov theorem and elementary proofs

Abstract

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.