# Dobrushin Uniqueness Theorem

The Dobrushin Uniqueness Theorem and Its. Path Coupling, Dobrushin Uniqueness, and Approximate Counting by R Bubley & M E Dyer January 1997. Now we formulate such a theorem. a-priori measure, can be extended, in a unique manner, to a full speciﬁca-tion. Applying Dobrushin's uniqueness theorem to these models, we are able to obtain the mean field result up to a constant numerical factor. For a good exposition of the initial theory. A Theorem on Color Coding,'' Memorandum 40-130-153, July 8, 1940, Bell Laboratories. principal eigenvalue For any transition measure p(t,x,dy) assiciated with L, there exist λ c so that L − λis subcritical for λ>λ. A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. existence and uniqueness theorem for (1. Dobrushin Uniqueness Techniques and the Decay of Correlations in Continuum Statistical Mechanics David Klein Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Abstract. 2, which as for the Dobrushin condition states that for an improved version of the interaction matrix being less then 1 in any metric norm implies uniqueness. Here we show that the problem (3. Let d 2 and q 4d(d+1). Dobrushin, O. Dobrushin's Uniqueness Theorem. A natural guess would be that uniqueness holds if and only if A-Ex(u), but wewere unable to prove this. The uniqueness part of Theorem 1. version of the Dobrushin uniqueness theorem, with some measurability assumptions dropped; † proof of decay of correlations in systems where the assumptions of The- orem 2. Some results are known for Gibbs fields in the Dobrushin uniqueness region and this is exploited in Jensen (1993). This may be viewed as an extension ofVasserstein’s convergence theorem to the case where the interaction has both a local and a global component. Could anybody be serious who wrote papers with titles like "The Absence of Phase Transition for Antiferromagnetic Potts Models Via the Dobrushin Uniqueness Theorem" and "New Lower Bounds on the Self-Avoiding-Walk Connective Constant"?. The Multivariate Empirical of Long Memory Processes Ichaou Mounirou * Faculty of Management and Economics Sciences, Université de Parakou, Benin *Corresponding Author: Mounirou Ichaou Faculty of Management and Economics Sciences, Université de Parakou, Benin E-mail: [email protected] Begin Gibbs measures. Wulff shape of crystals. 58 in the book. 3 Uniqueness in one dimension 164 Chapter 9 Absence of symmetry breaking. In Section 3 we present a theorem of LDT for Markov chain and results for hypotheses testing. We prove that in a given region of the (ß, µ) plane, where ß is the inverse temperature, and µ is the chemical potential, either the Gibbs state is unique or it does not exist. The Dobrushin uniqueness condition  guarantees that, in any dimension, disorder persists at su ciently high temperature (small ). LaTex2e, 80 pages including 4 figuresInternational audienceMotivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let \alpha be a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" \beta_h = I_{1-h} + I_h \alpha for h: X \to [0,1] (here I_f denotes the diagonal matrix with. He further showed that this is indeed a ''good" potential kernel in the sense that one can write the solution of certain equations in terms of this kernel, and he studied the. The Dobrushin Uniqueness Theorem and Its. The localisation of low-temperature interfaces in ddimensional Ising model Wei Zhou June 24, 2019 Abstract We study the Ising model in a box in Zd(not necessarily parallel to the directions of the lattice) with Dobrushin boundary conditions. 1 This led us to investigate the cleaning process in its own right. Theory of Probability & Its Applications 45:2, (1997) Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Starting with Dobrushin  in 1968, physicists have been developing techniques to characterize the regimes on either side of c for the hard-core model. He proved the famous Do-brushin Uniqueness Theorem in 1968, ensuring the uniqueness of the Gibbs mea-sures for a very general class of interactions at very high temperatures ( ˝1). on the Dobrushin uniqueness condition. The uniqueness criteria for general q-process were obtained by Chen and Zheng. Ising model I discuss here the basic theorems about existence and non-existence of phase transitions, in particular the "Peierls argument" and the "Dobrushin unique-ness theorem," which have fundamental importance in the whole book. brushin condition, see Theorem D. For more than half-century ago, some criteria for the uniqueness were known. 1 (Uniqueness Theorem) If the vector ﬁeld X(x ,t ) satisﬁes a Lipschitz condition in a domain R , then there is at most one solution x( t ) of the diﬀerential system. Therefore, the Gibbs measure applies to widespread problems outside of physics , such as Hopfield networks , Markov networks , Markov logic networks , and. Spectral gap for the zero range process with constant rate. Mark Ashbaugh (1980), Professor of Mathematics, University of Missouri, Columbia Thesis: Asymptotic perturbation theory for the eigenvalues of Schrödinger operators in a strong coupling limit. For Gibbs temperature states, the scheme of the proof of the noncommutative central limit theorem is given by using the commutative central limit theorem for corresponding Euclidean measures. Bainov and M. 3 Uniqueness in one dimension 164 Chapter 9 Absence of symmetry breaking. Outside the Dobrushin uniqueness region very little is known. The connection between the non-uniqueness. 2, which as for the Dobrushin condition states that for an improved version of the interaction matrix being less then 1 in any metric norm implies uniqueness. (1997) Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. USSR, Moscow. Dobrushin’s uniqueness condition, or be in the -high temperature regime. Suslin, "Existence and uniqueness theorems for an inhomogeneous kinetic equation with the Ühling-Uhlenbeck collision integral (the case of Fermi-Dirac statistics)", Preprint No. 2 is already known | see [KL07, Law09, PW17]. 1 Discrete symmetries in one dimension 169 9. existence and uniqueness theorem for (1. Ricci curvature of Markov chains on metric spaces Yann Ollivier Abstract We deﬁne the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. Dobrushin's condition was introduced byDobrushin(1968) in the study of Gibbs measures, originally in the context of identifying conditions under which the Gibbs distribution has a unique equilibrium / stationary state and has since been well-studied in statistical physics and probability. 1 Introduction Dobrushin's uniqueness theorem [1,2,3,4] provides a simple but powerful method for proving the uniqueness of the infinite-volume Gibbs measure, as well as the exponential decay of correlations in this unique Gibbs measure, for classical. Two Examples. Spectral gap for the zero range process with constant rate. In order to verify the local Dobrushin condition we use continuity (regularity) of the heat kernel combined with some weak form of irreducibility similarly to . 1, we have that the continuum Potts measure is unique for sufficiently small z (depending on φ, ψ and q). Let us now examine this theorem in detail. de la Rue : 2008 issue 1: A Functional Central Limit Theorem for Interacting Particle Systems on Transitive Graphs P. The Q-processes may not be unique in general, but there always exists the minimal one, due to Feller (1940, Theorem 1), denoted by Pmin(t) = pmin ij (t) : i;j2E. Shlosman: Absence of breakdown of continu-ous symmetry in two-dimensional models of statistical physics, Comm. Introduction 1. Haller and T. The Dobrushin Uniqueness Theorem and Its. main theorem is the following:. 1 Introduction. Pechersky, A criterion of the uniqueness of Gibbsian fields in the non-compact case, Lecture Notes Math. 2,that jjjT~jjj H = 1 min i 2r. We prove a partial uniqueness theorem for random-cluster measures, and make certain conjec-tures about uniqueness and translation-invariance. In the case of A1, for any distribution 0 2P1, there is a unique global weak solution (t) 2C([0;+1);P1) with initial condition 0 to (1. Dependence is introduced through the interactions potentials of equilibrium statistical mechanics. It was motivated by a desire to understand the obstructions to k-partitionability in the original random context. uniqueness of exp onen t is not ob vious form this de nition, although it is unique b y Theorem 1. The Hammersley-Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. We show that macroscopic con-vergence implies weak convergence ofthe underlying microscopic process{Xt}t∈N to the same limiting random eld. How to Clean a Dirty Floor: Probabilistic Potential Theory and the Dobrushin Uniqueness Theorem T. Theorem 3 allows us to obtain the uniqueness theorem of the DLR equations for the self-similar field constructed in Theorem 2. In Section 4 we improve this result for some common lattices. Doob Brownian Motion on a Green Space. Haller and T. This leads. This deﬁnition naturally extends to any Markov chain on a metric space. The problem was recently resolved by Chigansky and van Handel. the models we discuss are already known to admit a unique Gibbs measure by other methods, for most of them, our results extend the range of parameters for which uniqueness is established using "ﬁnite size" conditions of the Dobrushin type. The topic of uniqueness and the related theme of mixing properties of random fields was. When I wrote back to NPR I admitted that we should have known Sokal was putting us on. ShannonMacMillan theorems for random elds along curv es y Dobrushin Kotec ky and Shlosman suggests that suc is the unique represen tation of b yin tegers p N. EDU Eric Vigoda [email protected] Advances in Mathematical Physics is a peer-reviewed, Open Access journal that publishes original research articles as well as review articles that seek to understand the mathematical basis of physical phenomena, and solve problems in physics via mathematical approaches. Modern physical theories predict that near the critical point β = β cr the limiting Gibbs measure P β must be invariant with respect to renormalizations of the system. CRITICAL ISING ON THE SQUARE LATTICE MIXES IN POLYNOMIAL TIME EYAL LUBETZKY AND ALLAN SLY Abstract. on the Dobrushin uniqueness condition. 1021, Springer-Verlag , Berlin, New York, Heidelberg, 1982, pp. Lower bound for IID process level LDP. Zegarlinski B, 1992, Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, Journal of Functional Analysis, Vol:105, Pages:77-111 DOI Zegarlinski B, Stroock DW, 1992, The equivalence of the logarithmic Sobolev inequality and the Dobrushin - Shlosman mixing condition. ; High-temperature differentiability of lattice Gibbs states by Dobrushin uniqueness A unified Existence and Ergodic Theorem for Markov. A sto c hastic pro cess f X (t) g is said to ha v e stationary incremen ts, if the distributions of f X (h + t) g are indep enden tof. It also has various historical sources. On a unique solution of a differential equation in vector distribution, Michael V. Dobrushin's Approach to Queueing Network Theory 375 tionary ergodic sequence (which wealways assumebelow) and obey the non-overload condition Es0, R T 0 k R. 117-167 (1998). A natural guess would be that uniqueness holds if and only if A-Ex(u), but wewere unable to prove this. Ising model I discuss here the basic theorems about existence and non-existence of phase transitions, in particular the "Peierls argument" and the "Dobrushin unique-ness theorem," which have fundamental importance in the whole book. We give conditions for the existence and uniqueness of an invariant measure, and show the convergence in distribution. Dobrushin, “The description of a random field by means of conditional probabilities and conditions of its regularity. 1, we have that the continuum Potts measure is unique for sufficiently small z (depending on φ, ψ and q). Under this condition, he proved “unique-ness of the Gibbs measure,” which roughly stated says that there is asymptotically no correlation between the spin at a site v and the spins at sites at distance d from v, as d tends to inﬁnity. Journal of Statistical Physics 86 :3-4, 551-579. 34, Número 5. The essential ingredient in our analysis is an estimate of the Vaserstein distance between single-site Gibbs states corresponding to different exterior spin configurations. THEOREM EFS If there is a renormalized interaction for one translation-invariant Gibbs measure, then the renormalized measure of any other translation-invariant measure is a Gibbs measure for the same interaction. on the Dobrushin uniqueness condition. In higher dimensions, the famous result of Dobrushin in [Dob72] says that at low temperature, the interface in a straight box is localised around the middle hyperplane of the box when the temperature is low. 1021, Springer-Verlag , Berlin, New York, Heidelberg, 1982, pp. As a consequence of Dobrushin's theorem, if the degree of every vertex in Sleater, Tarjan, and Thurston  used a diﬀerent setup to count the number of ours appears to be more general as well as simpler. '' Not included. Memories of Roland Dobrushin Robert Minlos, Senya Shlosman, and Nikita Vvedenskaya 428 N OTICES OF THE AMS V OLUME 43, NUMBER 4 Roland Dobrushin died in Moscow on November 12, 1995, at the age of sixty-six.  Zegarlinski B. 00, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in. "Dobrushin Uniqueness Techniques," statistical mechanics graduate seminar, Indian Statistical Institute, Delhi, July-August 1982. random Gibbs measures of Theorems 2 and 3. This universality' theorem includes as special cases results of Aizenman and Lebowitz, Gravner and Gri eath, Mountford, and van Enter and Hulshof, signi cantly strengthens bounds of Bollob as, Smith and Uzzell, and complements recent work of Balister, Bollob as, Przykucki and Smith on subcritical models. Therefore, the Gibbs measure applies to widespread problems outside of physics , such as Hopfield networks , Markov networks , Markov logic networks , and. Oscillatory properties and asymptotic behavior of the solutions of a class of operator-differential equations. Isoperimetry in the in nite cluster Julian Gold UCLA Theorem (Biskup, Louidor, Procaccia, Rosenthal ’12) unique up to translations2. This item: A Course on Large Deviations With an Introduction to Gibbs Measures (Graduate Studies in Mathematics… by Firas Rassoul-agha Hardcover \$79. and Zagrebnov, Valentin A. A fundamental question is whether at low temperature (large ) the model remains disordered or, instead, undergoes a phase transition into an ordered phase. Firas Rassoul-Agha, University of Utah, Salt Lake City, UT and Timo Seppäläinen, University of Wisconsin-Madison, Madison, WI. Non-existence 168 9. In the follo wing, w e. A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density. 3) appeared ﬁrst in the work of Dobrushin (1970) with his study on random ﬁelds, and is known in statistical mechanics as the "Dobrushin uniqueness condition". As subsequent work would show [6, 8, 11, 1, 17], the same "Dobrushin condition" implies several other. Here we show that the problem (3. 105, 77-111 (1992) Zbl0761. Some results are known for Gibbs fields in the Dobrushin uniqueness region and this is exploited in Jensen (1993). A Theorem on Color Coding,'' Memorandum 40-130-153, July 8, 1940, Bell Laboratories. This note gives an interpretation of Dobrushin?s conditions for uniqueness and mixing Markov field in terms of a random walk with an absorbing state. Central Limit Theorem Laws of Large Numbers: weak and strong Toolbox: moment generating functions, Bernstein-Chernoff-Hoeffding bounding technique, Hoeffding bound for a. version of the Dobrushin uniqueness theorem, with some measurability assumptions dropped; † proof of decay of correlations in systems where the assumptions of The- orem 2. the mixing coefficients. Key Words: Dobrushin uniqueness theorem, antiferromagnetic Potts models, phase transition. In addition, an algebraic characterization is presented. Statistical analysis of uniformity trials. 4 The replica overlap 180 9. 2, which as for the Dobrushin condition states that for an improved version of the interaction matrix being less then 1 in any metric norm implies uniqueness. For a good exposition of the initial theory. The ones marked * may be different from the article in the profile. Outside the Dobrushin uniqueness region very little is known. Comparison Theorems for Gibbs Measures Comparison Theorems for Gibbs Measures Rebeschini, Patrick; Handel, Ramon 2014-08-08 00:00:00 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. main theorem is the following:. (1996) Coastline detection by a Markovian segmentation on SAR images. Dobrushin's condition of weak depen-dence, 268 dominated convergence theorem, 495 Doob Dynkinlemma, 493 dropletmodel, 216 dual lattice, 110 duality, 132 dusting lemma, 269 Dynkinsystem, 493 effective interface models, 407 energy density average, 300 average, Curie Weiss, 74 Curie Weiss, 64 ground state, 234 energy shell, 19 energy. Vigoda then introduced an alternative Markov. Section 6 contains a proof of the concen-tration estimates for expectations w. He brought it into the theory of random fields and developed the theory, which is now known as 'Dobrushin uniqueness'. Isoperimetry in the in nite cluster Julian Gold UCLA Theorem (Biskup, Louidor, Procaccia, Rosenthal ’12) unique up to translations2. lattice site are unique. The term abstract polymer system was coined in , but the mathematical theory of these systems goes. Ycart : 2008 issue 1. Linnik On the Decomposition of the Convolution of Gaussian and Poissonian Laws. Dobrushin began the study of non-uniqueness of the DLR Gibbs measure and proposed its interpretation as a phase transition. Tatikonda , Decay of correlation in network flow problems, in 2016 Annual Conference on Information Science and Systems (CISS), 2016, 169-174. Tetyana Pasurek for their patient guidance, encourage. Ships from and sold by Amazon. `A Theorem on Color Coding,'' Memorandum 40-130-153, July 8, 1940, Bell Laboratories. It is well-known (e. This theorem says that for every continuous function f on the real line and every positive continuous function e there exists a holomorphic function h, such that the difference between f and h is always smaller than e. ACKNOWLEDGEMENTS I would like to express my gratitude to my supervisor Prof. a-priori measure, can be extended, in a unique manner, to a full speciﬁca-tion. Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model Charilaos Efthymiou∗, Thomas P. Namely, for potentials satisfying a condition similar to the one by Bowen, we obtain in. Beyond the large deviations of independent variables and Gibbs measures, later parts of the book treat large deviations of Markov chains, the Gärtner-Ellis theorem, and. 00, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. Some results are known for Gibbs fields in the Dobrushin uniqueness region and this is exploited in Jensen (1993). the Dobrushin-Lanford-Ruelle (DLR) equations (see [5,6]) with the HM Hamiltonian with the external field h. We also prove non-existence of covariant or deterministically directed bi-inﬁnite polymer measures. We also consider the standard convention that : 0 log 0 = 0, , if a >0. the Dobrushin [7,8,10,11,13,18,21] and Dobrushin–Shlosman [1,9,22] uniqueness theorems. theorem and a large deviation principle. In Section 3 we prove that the Dobrushin uniqueness theorem is applicable to the q-state Potts antiferromagnet on a lattice of maximum coordination number r, uniformly at all temperatures (including zero temperature), whenever q > 2r. Thesis: The Dobrushin uniqueness theorem and its application to the classical N-vector models. are formulated as classical limit theorems in probability theory such as the law of large numbers, the central limit theorem and the large deviation principles. Notice that instead of the famous Yamada. without the Kac potential) satisfies a mixing condition at the temperature T, then, at the same temperature the full interaction (i.