Gibbs Ensembles of Nonintersecting Paths

We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices.     

Complete Equivalence of the Gibbs Ensembles for One-Dimensional Markov Systems

We show the equivalence of the Gibbs ensembles at the level of measures for one-dimensional Markov-Systems with arbitrary boundary conditions. That is, the limit of the microcanonical Gibbs ensemble is a Gibbs measure with an interaction depending on the microcanonical constraint. In fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins (including the boundary spins) are fixed. When conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble. We show that the limit is a Gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures. For this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions. We establish analogous results for finite range potentials.     

Gibbs Distributions for Random Partitions Generated by a Fragmentation Process

In this paper we study random partitions of {1,…,n} where every cluster of size j can be in any of w _j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w _j , the time-reversed process is the discrete Marcus–Lushnikov coalescent process with affine collision rate K _i,j = a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton–Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions. Research supported in part by N.S.F. Grant DMS-0405779.     

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the Hilbert-Schmidt ensemble. These universal limits have been proved before for determinantal Hermitian matrix ensembles and for some special classes of the Wigner random matrices. Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”. Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik,” and grants RFBR-05-01-00911, DFG-RFBR-04-01-04000, and NS-638.2008.1.     

On the Zero-Temperature Limit of Gibbs States

We exhibit Lipschitz (and hence Hölder) potentials on the full shift ${\{0,1\}^{\mathbb{N}}}We exhibit Lipschitz (and hence H?lder) potentials on the full shift {0,1}^\mathbbN{\{0,1\}^{\mathbb{N}}} such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are “exponentially decaying” interactions on the configuration space {0,1}^{\mathbb Z}{\{0,1\}^{\mathbb Z}} for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space {0,1}^\mathbbZd{\{0,1\}^{\mathbb{Z}^{d}}}, d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials.     

Limit Gibbs state for some classes of one-dimensional systems of quantum statistical mechanics

We prove the existence of the limit Gibbs state for one-dimensional continuous quantum fermion systems with non-hard-core, non-negative, rapidly decreasing pair interaction potentials. Existence of the limit Gibbs state is also established for one-dimensional continuous quantum boson systems with pair interaction potentials as above which, in addition, increase sufficiently fast at small distances.This work is partially performed with the support of the National Council of Researches of Italy     

Statistical Ensembles for Economic Networks

Economic networks share with other social networks the fundamental property of sparsity. It is well known that the maximum entropy techniques usually employed to estimate or simulate weighted networks produce unrealistic dense topologies. At the same time, strengths should not be neglected, since they are related to core economic variables like supply and demand. To overcome this limitation, the exponential Bosonic model has been previously extended in order to obtain ensembles where the average degree and strength sequences are simultaneously fixed (conditional geometric model). In this paper a new exponential model, which is the network equivalent of Boltzmann ideal systems, is introduced and then extended to the case of joint degree-strength constraints (conditional Poisson model). Finally, the fitness of these alternative models is tested against a number of networks. While the conditional geometric model generally provides a better goodness-of-fit in terms of log-likelihoods, the conditional Poisson model could nevertheless be preferred whenever it provides a higher similarity with original data. If we are interested instead only in topological properties, the simple Bernoulli model appears to be preferable to the correlated topologies of the two more complex models.     

Markov Partitions for dispersed billiards

Markov Partitions for some classes of billiards in two-dimensional domains on ² or two-dimensional torus are constructed. Using these partitions we represent the microcanonical distribution of the corresponding dynamical system in the form of a limit Gibbs state and investigate the character of its approximations by finite Markov chains.Dedicated to the memory of Rufus Bowen     

Matrix Kernels for Measures on Partitions

We consider the problem of computation of the correlation functions for the z-measures with the deformation (Jack) parameters 2 or 1/2. Such measures on partitions are originated from the representation theory of the infinite symmetric group, and in many ways are similar to the ensembles of Random Matrix Theory of β=4 or β=1 symmetry types. For a certain class of such measures we show that correlation functions can be represented as Pfaffians including 2×2 matrix valued kernels, and compute these kernels explicitly. We also give contour integral representations for correlation kernels of closely connected measures on partitions. Supported by US-Israel Binational Science Foundation (BSF) Grant No. 2006333.     

Conformal Radii for Conformal Loop Ensembles

The conformal loop ensembles CLE _κ , defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLE _κ gasket, which consists of points not surrounded by any loop, to be

Universality for Orthogonal and Symplectic Laguerre-Type Ensembles

We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K _{n, β}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (β=2) Laguerre-type ensembles have been proved by the fourth author in Ref. 23. The varying weight case at the hard spectral edge was analyzed in Ref. 13 for β=2: In this paper we do not consider varying weights. Our proof follows closely the work of the first two authors who showed in Refs. 7, 8 analogous results for Hermite-type ensembles. As in Refs. 7, 8 we use the version of the orthogonal polynomial method presented in Refs. 22, 25, to analyze the local eigenvalue statistics. The necessary asymptotic information on the Laguerre-type orthogonal polynomials is taken from Ref. 23.     

The Molecular-Dynamics Method for Different Statistical Ensembles

Modifications of the molecular-dynamics method for different statistical ensembles are examined. Particular emphasis is given to the Parrinello-Rahman method wherein the volume and shape of a molecular-dynamics cell are allowed to vary with time. The latter circumstance is of great importance because it enables processes involving marked structural changes in the system to be studied.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 16–23, February, 2005.     

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.     

Edge Universality for Orthogonal Ensembles of Random Matrices

We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of Shcherbina (Commun. Math. Phys. 285, 957–974, 2009) on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.     

Comparison Theorems for Gibbs Measures

The Dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high-dimensional probability distributions in terms of their local specifications. Originally introduced to prove uniqueness and decay of correlations of Gibbs measures, it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting Markov chains. However, the classical comparison theorem requires validity of the Dobrushin uniqueness criterion, essentially restricting its applicability in most models to a small subset of the natural parameter space. In this paper we develop generalized Dobrushin comparison theorems in terms of influences between blocks of sites, in the spirit of Dobrushin–Shlosman and Weitz, that substantially extend the range of applicability of the classical comparison theorem. Our proofs are based on the analysis of an associated family of Markov chains. We develop in detail an application of our main results to the analysis of sequential Monte Carlo algorithms for filtering in high dimension.     

Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.     

Compactness and the maximal Gibbs state for random Gibbs fields on a lattice

We prove for a general class of Gibbsian Random Field on that the set of tempered Gibbs states is compact. This class contains the Euclidean random fields. Moreover if the interaction is attractive, there is a unique minimal and maximal Gibbs state _– and ₊×_± are unique translation invariant ant and have the global Markov property. We also prove that uniqueness of the tempered Gibbs state is equivalent to the magnetizationsm _±=_±(q _x ) being equal which is true if the pressure is differentiable.     

Master Partitions for Large N Matrix Field Theories

We introduce a systematic approach for treating the large N limit of matrix field theories. Received: 30 October 1998 / Accepted: 7 March 1999     

Gram Matrices of Mixed-State Ensembles

International Journal of Theoretical Physics - Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in...