Gibbs Measures Research Articles

Three-state p-SOS models on binary Cayley trees

Abstract We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value 0 , 1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p > 0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p = 1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p = 1 do exhibit transitions between extremality and non-extremality.

Phase transition analysis of the Potts-SOS model with spin set {−1,0,+1} on the Cayley tree

Abstract Recently, the Potts-SOS model on Cayley trees was studied using the Kolmogorov consistency condition to investigate the existence and properties of Gibbs measures. The author previously examined the thermodynamic properties of the one-dimensional Potts-SOS model on the positive natural number lattice N , demonstrating the absence of phase transitions [Akin H, Phys. Scr. 99 (2024), 055 231]. In this study, we apply the cavity method to explore the non-uniqueness of Gibbs measures for the Potts-SOS model with spins {–1, 0, + 1} on a second-order semi-infinite Cayley tree. We address the phase transition problem by analyzing the model’s iterative equation system and perform stability analysis at fixed points to determine regions where phase transitions occur.

Universality laws for Gaussian mixtures in generalized linear models*

Abstract Let ( x i , y i ) i = 1 , … , n denote independent samples from a general mixture distribution ∑ c ∈ C ρ c P c x , and consider the hypothesis class of generalized linear models y ^ = F ( Θ ⊤ x ) . In this study, we investigate the asymptotic joint statistics of a family of generalized linear estimators ( Θ 1 , … , Θ M ) obtained either from (a) minimizing an empirical risk R ^ n ( Θ ; X , y ) or (b) sampling from the associated Gibbs measure exp ⁡ ( − β n R ^ n ( Θ ; X , y ) ) . Our main contribution is to characterize under which conditions the asymptotic joint statistics of this family depends (in a weak sense) only on the means and covariances of the class conditional features distribution P c x . In particular, this allows us to prove the universality of different quantities of interest, such as the training and generalization errors, redeeming a recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis. Finally, we discuss the applications of our results in different machine learning tasks of interest, such as ensembling and uncertainty quantification.

On the Superadditive Pressure for 1-Typical, One-Step, Matrix-Cocycle Potentials

Let (ΣT,σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma _T,\\sigma )$$\\end{document} be a subshift of finite type with primitive adjacency matrix T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T$$\\end{document}, ψ:ΣT→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi :\\Sigma _T \\rightarrow \\mathbb {R}$$\\end{document} a Hölder continuous potential, and A:ΣT→GLd(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}:\\Sigma _T \\rightarrow \ extrm{GL}_d(\\mathbb {R})$$\\end{document} a 1-typical, one-step cocycle. For t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} consider the sequences of potentials Φt=(φt,n)n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _t=(\\varphi _{t,n})_{n \\in \\mathbb {N}}$$\\end{document} defined by φt,n(x):=Snψ(x)+tlog‖An(x)‖,∀n∈N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\varphi _{t,n}(x):=S_n \\psi (x) + t\\log \\Vert \\mathcal {A}^n(x)\\Vert , \\, \\forall n \\in \\mathbb {N}.\\end{aligned}$$\\end{document}Using the family of transfer operators defined in this setting by Park and Piraino, for all t<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t<0$$\\end{document} sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials Φt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _t$$\\end{document}. This extends the results of the well-understood subadditive case where t≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 0$$\\end{document}. Prior to this, Gibbs-type measures were only known to exist for t<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t<0$$\\end{document} in the conformal, the reducible, the positive, or the dominated, planar settings, in which case they are Gibbs measures in the classical sense. We further prove that the topological pressure function t↦Ptop(Φt,σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\mapsto P_{\ extrm{top}}(\\Phi _t,\\sigma )$$\\end{document} is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.

On set of p-adic Gibbs measures for the countable state 1D SOS model

On set of p-adic Gibbs measures for the countable state 1D SOS model

Repulsion, chaos, and equilibrium in mixture models

Abstract Mixture models are commonly used in applications with heterogeneity and overdispersion in the population, as they allow the identification of subpopulations. In the Bayesian framework, this entails the specification of suitable prior distributions for the weights and locations of the mixture. Despite their popularity, the flexibility of these models often does not translate into the interpretability of the clusters. To overcome this issue, repulsive mixture models have been recently proposed. The basic idea is to include a repulsive term in the distribution of the atoms of the mixture, favouring mixture locations far apart. This approach induces well-separated clusters, aiding the interpretation of the results. However, these models are usually not easy to handle due to unknown normalizing constants. We exploit results from equilibrium statistical mechanics, where the molecular chaos hypothesis implies that nearby particles spread out over time. In particular, we exploit the connection between random matrix theory and statistical mechanics and propose a novel class of repulsive prior distributions based on Gibbs measures associated with joint distributions of eigenvalues of random matrices. The proposed framework greatly simplifies computations thanks to the availability of the normalizing constant in closed form. We investigate the theoretical properties and clustering performance of the proposed distributions.

Gibbs Dynamics for Fractional Nonlinear Schrödinger Equations with Weak Dispersion

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear fractional Schrödinger equation (FNLS) with initial data distributed via its associated Gibbs measure. We construct global strong solutions with the flow property for the FNLS on the support of the Gibbs measure in the full dispersive range, thus resolving a question proposed by Sun and Tzvetkov (Nonlinear Anal 213, paper no. 112530, 2021). As a byproduct, we prove the invariance of the Gibbs measure and almost sure global well-posedness for FNLS.

Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^{3}$

We show that within a C^{1} -neighborhood {\mathcal{U}} of the set of volume preserving Anosov diffeomorphisms on the three-torus {\mathbb{T}^3} which are strongly partially hyperbolic with expanding center, any f\in{\mathcal{U}}\cap{\operatorname{Diff}^2(\mathbb{T}^3)} satisfies the dichotomy: either the strong-stable and strong-unstable bundles E^{s} , E^{u} of f are jointly integrable, or any fully supported u -Gibbs measure of f is SRB.

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

An Inverse Cluster Expansion for the Chemical Potential

Interacting particle systems in a finite-volume in equilibrium are often described by a grand-canonical ensemble induced by the corresponding Hamiltonian, i.e. a finite-volume Gibbs measure. However, in practice, directly measuring this Hamiltonian is not possible, as such, methods need to be developed to calculate the Hamiltonian potentials from measurable data. In this work, we give an expansion of the chemical potential in terms of the correlation functions of such a system in the thermodynamic limit. This is a justification of a formal approach of Nettleton and Green from the 50’s, that can be seen as an inverse cluster expansion.

On positive fixed points of operator of Hammerstein type with degenerate kernel and Gibbs measures

On positive fixed points of operator of Hammerstein type with degenerate kernel and Gibbs measures

Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two

Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two

Translation invariant Gibbs measures for three state Hard-Core models in the case Hinge

Translation invariant Gibbs measures for three state Hard-Core models in the case Hinge

The $p$-adic Gibbs measure for the four-state stick-Hard-Core model on a Cayley tree

The $p$-adic Gibbs measure for the four-state stick-Hard-Core model on a Cayley tree

$(k_0^{(m)})$-periodic Gibbs measures for the fertile three-state Hard-Core model in the case Wand

$(k_0^{(m)})$-periodic Gibbs measures for the fertile three-state Hard-Core model in the case Wand

Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree

Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree

On Gibbs measures of the Ising Model on (k, m)-Ary Trees

On Gibbs measures of the Ising Model on <i>(k, m)</i>-Ary Trees

Central limit theorem in disordered Monomer‐Dimer model

AbstractWe consider the disordered monomer‐dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure with a rate of convergence. The central limit theorem continues to hold under a nearly optimal finite ‐moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan who showed a Gaussian central limit theorem in the disordered monomer‐dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer‐dimer model exhibits a decay of correlation with high probability. We also establish a central limit theorem for the Gibbs average of the number of dimers where the underlying graph has subexponential volume growth and the edge weights are Gaussians.

Perfect sampling of $q$-spin systems on $\mathbb{Z}^{2}$ via weak spatial mixing

We present a perfect marginal sampler of the unique Gibbs measure of a spin system on \mathbb{Z}^{2} . The algorithm is an adaptation of a previous “lazy depth-first” approach by the authors, but relaxes the requirement of strong spatial mixing to weak. Our result is a step towards methods of efficient sampling using only weak spatial mixing. The work exploits a classical result in statistical physics relating weak spatial mixing on \mathbb{Z}^{2} to strong spatial mixing on squares. When the spin system exhibits weak spatial mixing, the run-time of our sampler is linear in the size of sample. Applications of note are the ferromagnetic Potts model at supercritical temperatures and the ferromagnetic Ising model with consistent non-zero external field at any non-zero temperature.

Weakly periodic gibbs measures for the HC model with a countable set of spin values

Weakly periodic gibbs measures for the HC model with a countable set of spin values