On invariant generating sets for the cycle space

TL;DR

The paper investigates when a unimodular random graph or finitely generated Cayley graph admits an automorphism-invariant, locally finite generating set for its cycle space, a key step toward factor-of-iid constructions of the Loop $O(1)$ model via FK-Ising. It shows that geodesic cycles need not form such a generating set: on the lamplighter group $DL(2,2)$, for some $p_0<1$, both Bernoulli percolation and FK-Ising samples contain infinitely many geodesic cycles through the root with positive probability. The authors further prove that the property of having an invariant locally finite generating set is preserved under percolation when $p$ is near 1, enabling factor-of-iid realizations of the Loop $O(1)$ model on Cayley graphs of finitely presented groups. As a consequence, there exists a threshold $ar{x}<1$ such that the free Loop $O(1)$ model on such graphs is a factor of iid for all $x$ with $ar{x}<x esim 1$, with extensions to graphs whose cycle space is generated by cycles of bounded length; these results illuminate the limits of geodesic-cycle-based approaches and open avenues for percolation-based constructions of invariant generators.

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.

Figures (1)

• Figure 1: The two subtrees $T_{1,n}$ and $T_{2,n}$. We called $T_o$ the edges of $\omega_p$ whose first coordinate is in the left tree and second coordinate is in the $o_2$-$\hat{o}_2$ path on the right. It is distributed as Bernoulli($p$) percolation on the left tree, and thus will have some vertices ${\ell_1}\in L_1$ and ${\ell'_1}\in L_1'$ in the cluster of $o_1$ with probability at least $p^2\theta(p)^2$. Then we look at the $T_{{\ell_1}}\subset\omega_p$ of edges whose first coordinate is in the $o_1$-${\ell_1}$ path on the left and second coordinate is in the right tree. Define $T_{{\ell'_1}}$ similarly. With uniformly positive probability, the two independent Bernoulli($p$) configurations that arise if we project $T_{{\ell_1}}$ and $T_{{\ell'_1}}$ to the right tree, both contain some element $(o_1,x_2)$ with $x_2\in L_2$. A geodesic cycle in $\omega_p$ is then given by the open paths between the consecutive pairs $o$, $({\ell_1},\hat{o}_2)$, $(o_1,x_2)$, $({\ell'_1},\hat{o}_2)$, $o$.

Theorems & Definitions (7)

• Theorem 1
• Remark 2
• Theorem 3
• Theorem 4
• Theorem 5: ADTW,HJL,HSr
• Lemma 6
• Lemma 7