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Subject: Gibbs measures × WGL Institute: WIAS ×

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Now showing 1 - 7 of 7
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    Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures
    ([Madralin] : EMIS ELibEMS, 2018) Cotar, Codina; Jahnel, Benedikt; Külske, Christof
    The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.
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    Attractor properties for irreversible and reversible interacting particle systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Jahnel, Benedikt; Külske, Christof
    We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.
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    Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bolthausen, Erwin; König, Wolfgang; Mukherjee, Chiranjib
    We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]
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    Large deviations for empirical measures generated by Gibbs measures with singular energy functionals
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dupuis, Paul; Laschos, Vaios; Ramanan, Kavita
    We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on n-particle configurations, each of which is defined in terms of an inverse temperature bn and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (bn)/(n) ® ¥, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed bn =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in ``A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in ``First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.
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    Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Köppl, Jonas
    We consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.
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    Extremal decomposition for random Gibbs measures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Cotar, Codina; Jahnel, Benedikt; Külske, Christof
    The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.
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    Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) König, Wolfgang; Mukherjee, Chiranjib
    In this paper, we study a transformed path measure that arises from a mean-field type interaction of a three dimensional Brownian motion in a Coulomb potential. Under the influence of such a transformed measure, the large-t behavior of the normalized occupation measures, denoted by Lt, is of high interest. This is intimately connected to the well-known polaron problem from statistical mechanics and a full understanding of the behavior of Lt under the aforementioned transformation is crucial for the analysis of the polaron path measure under ‘strong coupling’ , its effective mass and justification of mean-field approximations. For physical relevance of this model, we refer to [S86]. Some mathematically rigorous research in this direction began in the 1980s with the analysis of the partition function of Donsker and Varadhan ([DV83-P]), but it was not until recently that a new technique was developed [MV14] for handling the actual path measures, and the main results the present paper, besides being interesting on their own, make determinant contribution towards a deeper analysis and a full identification of the limiting distribution of Lt under the transformed path measure.