On invariant generating sets for the cycle space

TL;DR

This paper investigates conditions under which the cycle space of unimodular random graphs or Cayley graphs admits an invariant, locally finite generating set, with implications for models like FK-Ising and free Loop O(1).

Contribution

It demonstrates that geodesic cycles do not always form such a generating set, providing a counterexample in the FK-Ising model on the lamplighter group, and explores the preservation of this property under Bernoulli percolation.

Findings

Geodesic cycles do not always generate the cycle space.

Counterexample in FK-Ising model on lamplighter group.

Invariant generating sets are preserved under Bernoulli percolation.

Abstract

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop $O(1)$ model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop $O(1)$ model.