Percolation on nonunimodular transitive graphs

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with $p_c=1$ on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ends in free minimal spanning forests

We show that for a transitive unimodular graph, the number of ends is the same for every tree of the free minimal spanning forest. This answers a question of Lyons, Peres and Schramm.Comment: Published at http://dx.doi.org/10.1214/009117906000000025 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Poisson allocation of optimal tail

The allocation problem for a $d$-dimensional Poisson point process is to find a way to partition the space to parts of equal size, and to assign the parts to the configuration points in a measurable, "deterministic" (equivariant) way. The goal is to make the diameter $R$ of the part assigned to a configuration point have fast decay. We present an algorithm for $d\geq3$ that achieves an $O(\operatorname {exp}(-cR^d))$ tail, which is optimal up to $c$. This improves the best previously known allocation rule, the gravitational allocation, which has an $\operatorname {exp}(-R^{1+o(1)})$ tail. The construction is based on the Ajtai-Koml\'{o}s-Tusn\'{a}dy algorithm and uses the Gale-Shapley-Hoffman-Holroyd-Peres stable marriage scheme (as applied to allocation problems).Comment: Published at http://dx.doi.org/10.1214/15-AOP1001 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On invariant generating sets for the cycle space

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

Factor of iid's through stochastic domination

We develop a method to prove that certain percolation processes on amenable random rooted graphs are factors of iid (fiid), given that the process is a monotone limit of random finite subgraphs that satisfy a certain independent stochastic domination property. Among the consequences are the previously open claims that the Uniform Spanning Forest (USF) is a factor of iid for recurrent graphs, it is a finite-valued finitary fiid on amenable graphs, and that the critical Ising model on $\Z^d$ is a finite-valued finitary fiid, using the known uniqueness of the Gibbs measure

Controllability, matching ratio and graph convergence

There is an important parameter in control theory which is closely related to the directed matching ratio of the network, as shown by Liu, Slotine and Barab\'asi (2011). We give proofs on two main statements of that paper on the directed matching ratio, which were based on numerical results and heuristics from statistical physics. First, we show that the directed matching ratio of directed random networks given by a fix sequence of degrees is concentrated around its mean. We also examine the convergence of the (directed) matching ratio of a random (directed) graph sequence that converges in the local weak sense, and generalize the result of Elek and Lippner (2009). We prove that the mean of the directed matching ratio converges to the properly defined matching ratio parameter of the limiting graph. We further show the almost sure convergence of the matching ratios for the most widely used families of scale-free networks, which was the main motivation of Liu, Slotine and Barab\'asi.Comment: 22 page

A comprehensive characterization of Property A and almost finiteness

Three important properties of groups, amenability, Property A and almost finiteness, are in our focus, in the wider context of general countable bounded degree graphs. A graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on the boundary of a tile is uniformly small. We give various equivalents for Property A and for strong almost finiteness. In particular, we prove that Property A together with a uniform version of amenability is equivalent to strong almost finiteness. Using these characterizations, we show that graphs of subexponential growth and Schreier graphs of amenable groups are always strongly almost finite, generalizing the celebrated result of Downarowicz, Huczek and Zhang about amenable Cayley graphs, based on graph theoretic rather than group theoretic principles. We also show that if a sequence of graphs of Property A (in a uniform sense) converges to a graph $G$ in the neighborhood distance (a purely combinatorial analogue of the classical Benjamini-Schramm distance), then their Laplacian spectra converge to the Laplacian spectrum of $G$ in the Hausdorff distance. Finally, we apply the previous theory to construct a new and rich class of classifiable $C^{\star}$-algebras. Namely, we show that for any minimal strongly almost finite graph $G$ there are naturally associated simple, nuclear, stably finite $C^{\star}$-algebras that are classifiable by their Elliott invariants.Comment: 44 pages, 1 figur

Finite-energy infinite clusters without anchored expansion

Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(G), the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.Comment: 9 pages, 1 figure. Small changes throughout. To appear in Bernoull