Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton-Tarjan √ n separation result for planar graphs. Connections to relaxed versions of quasi-isometries are explored, such as regular and semiregular maps. * Oded died while solo climbing Guye Peak in Washington State on September 1, 2008.

We show that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold.

Abstract. We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of Z d , where "connectedness" and "boundary" are understood with respect to various graphs on the vertices of Z d . These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory instead of topology. We generalize these results about Z d and Z d * to a very general family of pairs of graphs; see Lemma 2, Theorem 3 and Theorem 4. Our method also gives an elementary and short alternative to the original proofs for the cubic grid case. This approach seems to be efficient to treat possible other questions about the connectedness of boundaries. Although [K] mentions that some use of algebraic topology seems to be unavoidable, the greater generality (and simplicity) of our proof is a result of using purely graph-theoretic arguments. Also, it makes slight modifications of the results (such as considering boundaries in some subset of Z d instead of boundaries in Z d ) straightforward, while previously one had to go through the original proofs and make significant modifications. Denote byIn two dimensions, the use of some duality argument makes connectedness of boundaries more straightforward to prove. The lack of duality (that is, the correspondance that a cycle in one graph is a separating set in its dual) in higher

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

We study isomorphism invariant point processes of R d whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, a mapping of the point configuration to a graph on it that is measurable and equivariant with the point process. This answers a question of Holroyd and Peres. The tree will be used to construct a factor isomorphic to Z n . This perhaps surprising result (that any d and n works) solves a problem by Steve Evans. The construction, based on a connected clumping with 2 i vertices in each clump of the i'th partition, can be used to define various other factors.

Consider an ergodic unimodular random one-ended planar graph G of finite expected degree. We prove that it has an isometry-invariant embedding in the euclidean plane with no accumulation points if and only if it is invariantly amenable.By "no accumulation points" we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. We define a unimodular embedding of G as one that is jointly unimodular with G when viewed as a decoration, and show that G has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings. The results have applications to percolation phase transitions.

Consider an ergodic unimodular random one-ended planar graph G of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By "locally finite" we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of G is one that is jointly unimodular with G when viewed as a decoration. We show that G has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.

We investigate generalisations of the classical percolation critical probabilities p c , p T and the critical probabilityp c defined by Duminil-Copin and Tassion [11] to bounded degree unimodular random graphs. We further examine Schramm's conjecture in the case of unimodular random graphs: does p c