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83
The Markov chain Monte Carlo method: an approach to approximate counting and integration. in Approximation Algorithms for NPhard Problems, D.S.Hochbaum ed
, 1996
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The multivariate Tutte polynomial (alias Potts model) for graphs and matroids.
 In Surveys in combinatorics
, 2005
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Uniform spanning forests
 ANN. PROBAB
, 2001
"... We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d ..."
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Cited by 89 (23 self)
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We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d � 4. In the present work, we extend Pemantle’s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation, and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: • The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. • The tail σfields of the WSF and the FSF are trivial on any graph. • On any Cayley graph that is not a finite extension of Z, all component trees of the WSF have one end; this is new in Z d for d � 5. • On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. • The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. • A Cayley graph is amenable iff for all ɛ> 0, the union of the WSF and Bernoulli percolation with parameter ɛ is connected. • Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees. We also present numerous open problems and conjectures.
The RandomCluster Model
, 2008
"... The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of elec ..."
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Cited by 68 (21 self)
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The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
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 55 (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
Relationship privacy: Output perturbation for queries with joins
 In ACM Symposium on Principles of Database Systems, 2009. [13] Yossi
"... We study privacypreserving query answering over data containing relationships. A social network is a prime example of such data, where the nodes represent individuals and edges represent relationships. Nearly all interesting queries over social networks involve joins, and for such queries, existing ..."
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Cited by 52 (8 self)
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We study privacypreserving query answering over data containing relationships. A social network is a prime example of such data, where the nodes represent individuals and edges represent relationships. Nearly all interesting queries over social networks involve joins, and for such queries, existing output perturbation algorithms severely distort query answers. We propose an algorithm that significantly improves utility over competing techniques, typically reducing the error bound from polynomial in the number of nodes to polylogarithmic. The algorithm is, to the best of our knowledge, the first to answer such queries with acceptable accuracy, even for worstcase inputs. The improved utility is achieved by relaxing the privacy condition. Instead of ensuring strict differential privacy, we guarantee a weaker (but still quite practical) condition based on adversarial privacy. To explain precisely the nature of our relaxation in privacy, we provide a new result that characterizes the relationship between ǫindistinguishability (a variant of the differential privacy definition) and adversarial privacy, which is of independent interest: an algorithm is ǫindistinguishable iff it is private for a particular class of adversaries (defined precisely herein). Our perturbation algorithm guarantees privacy against adversaries in this class whose prior distribution is numerically bounded.
Negative dependence and the geometry of polynomials
, 2008
"... We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measur ..."
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Cited by 46 (13 self)
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We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures and we construct counterexamples to several conjectures of Pemantle
Markov Chains and Polynomial time Algorithms
, 1994
"... This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing ..."
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Cited by 43 (0 self)
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This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.
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.