Results 1  10
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15
Stationary determinantal processes: phase multiplicity
 Bernoullicity, entropy, and domination, Duke Math. Journal
, 2003
"... We study a class of stationary processes indexed by Z d that are defined via minors of ddimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Pha ..."
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Cited by 20 (6 self)
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We study a class of stationary processes indexed by Z d that are defined via minors of ddimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong Kproperty, a particular strengthening of the usual K (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to independent identically distributed (i.i.d.) processes in the sense of ergodic theory). We obtain estimates of their entropies, and we relate these processes via stochastic domination
Determinantal Processes and . . .
, 2006
"... We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points i ..."
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Cited by 12 (1 self)
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We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L 2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.
Determinantal processes and independence
 Probab. Surv
"... We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in ..."
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Cited by 12 (1 self)
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We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions. Figure 1: Samples of translation invariant point processes in the plane: Poisson (left), determinantal (center) and permanental for K(z, w) = 1 πezw−1 2 (z2+w  2). Determinantal processes exhibit repulsion, while permanental processes exhibit clumping. 1 1
Affine systems: asymptotics at infinity for fractal measures
 Acta Appl. Math
"... Abstract. We study measures on R d which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and c ..."
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Cited by 4 (2 self)
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Abstract. We study measures on R d which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures µ induced by a finite family of affine mappings in R d (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure µ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when µ is singular carry a gradation, ranging from Cantorlike fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of µ for paths at infinity in R d. We show how properties of µ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in R d, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.
Change Intolerance in Spanning Forests
 J. Theoret. Probab
, 2001
"... . Call a percolation process on edges of a graph change intolerant if the status of each edge is almost surely determined by the status of the other edges. We give necessary and sucient conditions for change intolerance of the wired spanning forest when the underlying graph is a spherically symme ..."
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Cited by 2 (2 self)
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. Call a percolation process on edges of a graph change intolerant if the status of each edge is almost surely determined by the status of the other edges. We give necessary and sucient conditions for change intolerance of the wired spanning forest when the underlying graph is a spherically symmetric tree. x1.
Random complexes and ℓ 2 Betti numbers
 In preparation
, 2003
"... Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a wellstudied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2Betti number of the group. Our main aim, however, is to present the basic elements of a higherdimensional an ..."
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Cited by 2 (2 self)
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Abstract. Uniform spanning trees on finite graphs and their analogues on infinite graphs are a wellstudied area. On a Cayley graph of a group, we show that they are related to the first ℓ 2Betti number of the group. Our main aim, however, is to present the basic elements of a higherdimensional analogue on finite and infinite CWcomplexes, which relate to the higher ℓ 2Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin. §1. Introduction. Enumeration of spanning trees in graphs began with Kirchhoff (1847). Cayley (1889) evaluated this number in the special case of a complete graph. Cayley’s theorem was
Couplings of uniform spanning forests
 Proc. Amer. Math. Soc
"... We prove the existence of an automorphisminvariant coupling for the wired and the free uniform spanning forests on connected graphs with residually amenable automorphism groups. ..."
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Cited by 2 (1 self)
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We prove the existence of an automorphisminvariant coupling for the wired and the free uniform spanning forests on connected graphs with residually amenable automorphism groups.
MARKOV LOOPS, DETERMINANTS AND GAUSSIAN FIELDS
, 2007
"... The purpose of this note is to explore some simple relations between loop measures, spanning trees, determinants, and Gaussian Markov fields. These relations are related to Dynkin’s isomorphism (cf [1], [11], [7]). Their potential interest could be suggested by noting that loop measures were defined ..."
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Cited by 1 (0 self)
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The purpose of this note is to explore some simple relations between loop measures, spanning trees, determinants, and Gaussian Markov fields. These relations are related to Dynkin’s isomorphism (cf [1], [11], [7]). Their potential interest could be suggested by noting that loop measures were defined
Zero dissipation limit in the Abelian sandpile model
, 906
"... Abstract: We study the abelian avalanche model, an analogue of the abelian sandpile model with continuous heights, which allows for arbitrary small values of dissipation. We prove that for nonzero dissipation, the infinite volume limit of the stationary measures of the abelian avalanche model exist ..."
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Abstract: We study the abelian avalanche model, an analogue of the abelian sandpile model with continuous heights, which allows for arbitrary small values of dissipation. We prove that for nonzero dissipation, the infinite volume limit of the stationary measures of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. Moreover we obtain exponential decay of spatial covariances of local observables in the nonzero dissipation regime. We then study the zero dissipation limit and prove that the selforganized critical model is recovered, both for the stationary measures and for the dynamics. Keywords: Abelian avalanche model, burning algorithm, weighted spanning trees, Wilson’s algorithm, zerodissipation limit, selforganized criticality. 1