Liming Wu
In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett's $M-ε$ theorem for interacting particle systems.
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We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity $z$ and inverse temperature $β$. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions.
We give a detailed and refined proof of the Dobrushin-Pechersky uniqueness criterion extended to the case of Gibbs fields on general graphs and single-spin spaces, which in particular need not be locally compact. The exponential decay of correlations under the uniqueness condition has also been established.
The pairwise influence matrix of Dobrushin has long been used as an analytical tool to bound the rate of convergence of Gibbs sampling. In this work, we use Dobrushin influence as the basis of a practical tool to certify and efficiently improve the quality of a discrete Gibbs sampler. Our Dobrushin-optimized Gibbs samplers (DoGS) offer customized variable selection orders for a given sampling budget and variable subset of interest, explicit bounds on total variation distance to stationarity, and certifiable improvements over the standard systematic and uniform random scan Gibbs samplers. In ou…
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 pa…
We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail $σ$-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory and mixing propert…
Existence of generalized Busemann functions and Gibbs measures for random walks in random potentials
该研究在极弱假设下，证明了一类含有限可允许步长的格点随机势场随机游走的广义Busemann函数与Gibbs-Dobrushin-Landford-Ruelle测度的存在性，覆盖多类相关模型且适用于所有维度。
We establish formulae for the derivative of the Poincaré constant of Gibbs measures on both compact domains and all of $\R^d$. As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the Poincaré constant is strictly decreasing in the inverse temperature, and vice versa. Applying this result to the $O(2)$ model allows us to give a sharpened upper bound on its Poincaré constant. We further show that this model exhibits a qualitatively different zero-temperature behavior of the Poincaré and Log-Sobolev constants.
In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sob…
We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function lim…
We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1 (1995) \cite{MPS}]. We complement previous results on the existence of phase transitions at low temperature [González-Navarrete et al., J. Stat. Phys. 162 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [Dobrushin, Theory Probab. Appl. 13 (1968)], (2) Disagreement percolation [van …
We investigate unbounded continuous spin-systems with infinite-range interactions. We develop a new technique for deducing decay of correlations from a uniform Poincaré inequality based on a directional Poincaré inequality, which we derive through an averaging procedure. We show that this decay of correlations is equivalent to the Dobrushin-Shlosman mixing condition. With this, we also state and provide a partial answer to a conjecture regarding the relationship between the relaxation rates of non-ferromagnetic and ferromagnetic systems. Finally, we show that for a symmetric, ferromagnetic sys…
We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture $\frac12(μ^\pm+μ^\mp)$ of two reflection-symmetric Dobrushin states of the 3-dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finite-volume measures with deterministic boundary conditions. Finally we point out what the issues are in order to extend the analysis to the Potts model, and give a few conjectures.
We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by …
In this paper, we detail and complete the existing characterizations of the decimation of the Ising model on $\Z^2$ in the generalized Gibbs context. We first recall a few features of the Dobrushin program of restoration of Gibbsianness and present the construction of global specifications consistent with the extremal decimated measures. We use them to consider these renormalized measures as almost Gibbsian measures and to precise its convex set of DLR measures. We also recall the weakly Gibbsian description and complete it using a potential that admits a quenched correlation decay, i.e. a wel…
Weak Poincaré inequalities for two-component Deterministic-scan Metropolis-within-Gibbs samplers
本文通过弱Poincaré不等式分析了确定性扫描Metropolis-within-Gibbs采样器的收敛性质，并建立了相关理论结果及应用。
We consider Gibbs measures on the configuration space $S^{\mathbb{Z}^d}$, where mostly $d\geq 2$ and $S$ is a finite set. We start by a short review on concentration inequalities for Gibbs measures. In the Dobrushin uniqueness regime, we have a Gaussian concentration bound, whereas in the Ising model (and related models) at sufficiently low temperature, we control all moments and have a stretched-exponential concentration bound. We then give several applications of these inequalities whereby we obtain various new results. Amongst these applications, we get bounds on the speed of convergence of…
Dobrushin and Tirozzi [14] showed that, for a Gibbs measure with the finite-range potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino, Capocaccia, and Tirozzi [7] extended this result for a family of Gibbs measures for long-range pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for long-range pair potentials not satisfying the conditions given in [7], that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central…
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The mai…
Gibbs measure over the cone of vector-valued discrete measures
本文在向量值测度锥上定义了具有位置和速度特征的粒子气体的 Gibbs 测度，通过引入概率测度族和局部 Hamiltonian，将其定义为 DLR 方程的解，并证明 tempered Gibbs 测度子集非空且相对紧以确立其存在性。
In this paper we establish some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean field particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinski's theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by means of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of th…
This note, mostly expository, is devoted to Poincar{é} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact …
DLR-KMS correspondence on lattice spin systems
针对取值于紧辛流形的经典晶格自旋系统，本文证明无限体积 Gibbs 态与 KMS 态一一对应，从而确立 DLR 条件与经典 KMS 条件等价。
We consider - in uniformly strictly convex potential regime - two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters though the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension $d = 2$, while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradien…
Mixing Time Bounds for the Gibbs Sampler under Isoperimetry
本文基于等周性条件，推导了系统扫描和随机扫描基尔丁采样器的导电性界，扩展了对基尔丁采样收敛行为的理解，并展示了结果对log-Lipschitz和log-smooth目标分布的有效性。
Weak Poincaré Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model
研究通过先证明弱泛函不等式对逼近随机定位过程的正则分布序列成立，再转移到所需分布，证明了\(\beta < \frac{1}{2}\)时谢林顿 - 柯克帕特里克模型吉布斯测度的弱庞加莱不等式，使热启动格劳伯动力学可有效抽样，向解决相关猜想迈进。
Liquid-Gas phase transition for Gibbs point process with Quermass interaction
本文研究了具有克尔马斯相互作用的吉布斯点过程在低温度下的液-气相变存在性，证明了临界点处压力的非可导性及测度的非唯一性。
Simulated annealing and Weak Poincaré inequalities with inverse-polynomial accuracy for the Sherrington-Kirkpatrick model
针对Sherrington-Kirkpatrick模型的Glauber动力学采样，提出基于弱Poincaré不等式的多项式时间模拟退火算法，实现逆多项式精度，并绕过Talagrand开放问题获得配分函数估计。
Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials
本文证明梯度流可优化性与低温吉布斯测度的庞加莱及对数索博列夫不等式之间的等价联系，并据此给出非对数凹密度的新高效采样方法。
On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics
该研究基于已有框架推导欠阻尼朗之万动力学$L^2$距离下的指数衰减显式估计，证明庞加莱型不等式，得到更简洁的速率表达式，且凸势下优化摩擦系数后速率快于过阻尼情形。
Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian Free Field
该研究为一类满足特定条件的非凸势函数梯度界面模型建立 Helffer-Sjöstrand 表示，证明其标度极限为高斯自由场、协方差衰减速率与 GFF 一致，并推广相关不等式，为非凸势函数模型的普适性研究提供了新方法。
Weak Poincaré Inequalities, Simulated Annealing, and Sampling from Spherical Spin Glasses
本文提出基于弱庞加莱不等式的模拟退火分析框架，并利用局部到全局原则证明该不等式，首次实现球面自旋玻璃在超越唯一性阈值情形下的马尔可夫链采样保证。
Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscal…
A contiguity approach to replica symmetric marginals
本文提出概率腔邻接框架，以高温谢林顿-柯克帕特里克模型为例证明平均场吉布斯系统局部边际的复制对称收敛性，该方法可推广至其他平均场吉布斯系统。
In the context of interacting particle systems, we study the influence of the action of the semigroup on the concentration property of Lipschitz functions. As an application, this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems. We illustrate our approach in a variety of examples for which we obtain several new results with short and non-technical proofs. These examples include the symmetric and asymmetric exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics"). We also give a new proof of the Poincaré inequality, base…
Uniform logarithmic Sobolev inequalities for the 2D Coulomb gas at the diffusive temperature scale
该研究针对扩散温度尺度下的二维库仑气体系综，证明其满足与N无关的一致对数索伯列夫不等式，还推导了加权平面高斯测度的庞加莱与对数索伯列夫不等式，常数仅依赖总方差和总指数。
Sharp hypocoercive convergence estimates for underdamped Langevin dynamics with specular reflection
本文针对镜面反射的欠阻尼Langevin动力学，证明了在Poincaré不等式等条件下L^2指数收敛，速率按√m缩放，实现约束采样的平方根加速，并扩展到非凸域。
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