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89
Lectures on random planar curves and SchrammLoewner evolutions
, 2003
"... The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilist ..."
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Cited by 143 (7 self)
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The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilists and I therefore assume knowledge in stochastic calculus (Itô’s formula etc.), but no special background in basic complex analysis. These lecture notes are neither a book nor a compilation of research papers. While preparing them, I realized that it was hopeless to present all the recent results on this subject, or even to give the complete detailed proofs of a selected portion of them. Maybe this will disappoint part of the audience but the main goal of these lectures will be to try to transmit some ideas and heuristics. As a reader/part of an audience, I often think that omitting details is dangerous, and that ideas are sometimes better understood when the complete proofs are given, but in the present case, partly because the technicalities often use complex analysis tools that the audience might not be so
Processes on unimodular random networks
, 2007
"... We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability fr ..."
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Cited by 124 (6 self)
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We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasitransitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 56 (5 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
Percolation in the Hyperbolic Plane
, 2000
"... Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation ..."
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Cited by 47 (4 self)
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Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
A measurablegrouptheoretic solution to von Neumann’s problem
, 2007
"... We give a positive answer, in the measurablegrouptheory context, to von Neumann’s problem of knowing whether a nonamenable countable discrete group contains a noncyclic free subgroup. We also get an embedding result of the freegroup von Neumann factor into restricted wreath product factors. ..."
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Cited by 45 (4 self)
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We give a positive answer, in the measurablegrouptheory context, to von Neumann’s problem of knowing whether a nonamenable countable discrete group contains a noncyclic free subgroup. We also get an embedding result of the freegroup von Neumann factor into restricted wreath product factors.
The diameter of longrange percolation clusters on finite cycles
 Random Struct. Alg
, 2001
"... Bounds for the diameter and expansion of the graphs created by longrange percolation on the cycle Z/NZ, are given. 1 ..."
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Cited by 43 (8 self)
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Bounds for the diameter and expansion of the graphs created by longrange percolation on the cycle Z/NZ, are given. 1
Scaling limits for minimal and random spanning trees in two dimensions
, 1998
"... A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), a ..."
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Cited by 41 (8 self)
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A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in R 2 with density δ −2. A continuum limit is expressed through a consistent collection of trees (made of curves) which includes a spanning tree for every finite set of points in the plane. Sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for δ> 0), ii) the tree branches are given by curves which are regular in the sense of Hölder continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of R², of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in R 2, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scaleinvariant power bounds on the probabilities of repeated crossings of annuli.
Determinantal probability measures
, 2002
"... Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationships with ma ..."
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Cited by 38 (4 self)
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Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology,