Gibbs Measures Research Articles

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

Fourier transform and expanding maps on Cantor sets

abstract: We study the Fourier transforms $\widehat{\mu}(\xi)$ of non-atomic Gibbs measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehat{\mu}(\xi)\to 0$ at a polynomial rate as $|\xi|\to\infty$.

$$H_A$$-Weakly Periodic $$p$$-Adic Generalized Gibbs Measures for the $$p$$-Adic Ising Model on the Cayley Tree of Order Two

$$H_A$$-Weakly Periodic $$p$$-Adic Generalized Gibbs Measures for the $$p$$-Adic Ising Model on the Cayley Tree of Order Two

Robust self-assembly of nonconvex shapes in two dimensions.

We present fast simulation methods for the self-assembly of complex shapes in two dimensions. The shapes are modeled via a general boundary curve and interact via a standard volume term promoting overlap and an interpenetration penalty. To efficiently realize the Gibbs measure on the space of possible configurations we employ the hybrid Monte Carlo algorithm together with a careful use of signed distance functions for energy evaluation. Motivated by the self-assembly of identical coat proteins of the tobacco mosaic virus which assemble into a helical shell, we design a nonconvex two-dimensional model shape and demonstrate its robust self-assembly into a unique final state. Our numerical experiments reveal certain essential prerequisites for this self-assembly process: blocking and matching (i.e., local repulsion and attraction) of different parts of the boundary, and nonconvexity and handedness of the shape.

Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity

We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu+(−Δ)α2u=2γβeβ|u|2u on d-dimensional compact Riemannian manifolds Md, for a dispersion parameter α>d, some coupling constant β>0, and γ≠0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ>0, the measure is well-defined in the whole regime α>d and β>0 (Theorem 1.1 (i)), while in the focusing case γ<0 its partition function is always infinite for any α>d and β>0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure, in the defocusing case γ>0, in the whole regime α>d and β>0 (Theorem 1.4 (i)). In the large dispersion regime α>2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β>0 and γ≠0. Using the Gibbs measure, we prove that solutions are almost surely global for any β,γ>0, and that the Gibbs measure is invariant (Theorem 1.4 (ii)). (iii) Finally, in the particular case d=1 and M=T, we are able to exploit some probabilistic multilinear smoothing effects to build a global probabilistic flow for (expNLS) for 1+22<α≤2 for any β,γ>0 (Theorem 1.6).

Stochastic process generated by 1-D Ising model with competing interactions

We consider a stochastic process generated by 1-D Ising model with competing interactions and describe all distributions of this process. It is shown that the set of all limit Gibbs measures, i.e. phase diagram, consist of ferromagnetic, anti-ferromagnetic, paramagnetic and modulated phases. Also it is proven that on the set of ferromagnetic phases one can reach the phase transition.

Gibbs measures for hardcore-solid-on-solid models on Cayley trees

We investigate the finite-state p-solid-on-solid (p-SOS) model for p=∞ on Cayley trees of order k⩾2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1→3→5→6→7 . This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p > 0 and, in the case of k = 2, we demonstrate that, on the level of the functional equations, the transition p→∞ is continuous.

Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

ABSTRACT We consider fertile three-state Hard-Core (HC) models on a Cayley tree. It is known that there exist four types of such models: ‘wrench’, ‘wand’, ‘hinge’ and ‘pipe’. In cases ‘wand’ and ‘hinge’ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The concept of alternative Gibbs measure is introduced and in the case ‘wand’ translational invariance conditions for alternative Gibbs measures are found. Also, we show the existence of alternative Gibbs measures which are not translation-invariant.

Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree

Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree

Translation-invariant Gibbs measures for the Ising-Potts model on a second-order Cayley tree

Рассматривается модель смешанного типа - модель Изинга-Поттса с тремя состояниями на дереве Кэли. Получен критерий существования предельных мер Гиббса для этой модели на дереве Кэли произвольного порядка. Изучаются трансляционно-инвариантные меры Гиббса на дереве Кэли второго порядка. Доказано существование фазового перехода, т. е. найдена область значений параметра, при которых существуют от одной до семи мер Гиббса для модели Изинга-Поттса с тремя состояниями.

Gibbs measures for fertile models with hard-core interactions and four states

Gibbs measures for fertile models with hard-core interactions and four states

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

The qualitative properties of 1D mixed-type Potts-SOS model with 1-spin and its dynamical behavior

In this investigation, we consider the one-dimensional (1D) mixed-type Potts-SOS model, where the spin is within the range of {−1, 0, 1}. We elaborate thermodynamic characteristics of 1D Potts-SOS model through the application of three distinct analytical approaches. We provide a brief overview of all translation-invariant splitting Gibbs measures (TISGMs) applicable to this model. For the model with a boundary field condition, we provide a comprehensive analysis of the uniqueness and non-uniqueness properties of the subset of fully homogeneous splitting Gibbs masures (SGMs). Our demonstration reveals that the model under consideration does not exhibit a phase transition phenomenon. We are also curious in the stability study of the suggested fixed points associated with the Gibbs measures. We show that the magnetization decreases to zero. By means of the transfer matrix method, we compute the free energy, entropy and internal energy of the model.

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Mass transference principle are tools designed to provide estimates for the Hausdorff dimension of points approximable at a certain rate by a specific sequence of points (for instance rationals). Usually, such results are established provided that ambiant space is equipped with an Ahlfors regular measure (see [7,28] for instance). In the case of balls, Barral and Seuret established that mass transference principles still holds provided that the ambiant measure is a Gibbs measure or a self-similar measure satisfying the open set condition ([3]). In this article, we establish a mass transference principle “from ball to rectangle” assuming that the ambiant measure is the projection of Gibbs measure on the dyadic grid. Such measures are not Ahlfors regular in general and this result extends in particular the mass transference principle from ball to rectangle in the case of the Lebesgue measure, established in [28], and partially the result of Barral and Seuret [3]. As an application, given b∈N, we estimate the dimension of weighted approximation of points of [0,1]2 approximable by points with rational coordinates whose expansion in base b has prescribed frequencies. These results are new and surprising in the sens that, although the Lebesgue measure “scales” naturally rectangles, this needs not be the case for projection of Gibbs measures.

A Ray–Knight theorem for ∇ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\phi $$\\end{document} interface models and scaling limits

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which—generically—is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray–Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R^3$$\\end{document} with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p &gt; 1 p&gt;1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p &gt; 1 p&gt;1 and scattering for p &gt; 3 p&gt;3 (actually even for 1 &gt; p ≤ 3 1&gt;p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

A NEW CLASS OF GIBBS MEASURES FOR THREE-STATE SOS MODEL ON A CAYLEY TREE

The phase transition phenomenon is one of the central problems of statistical mechanics. It occurs when the model possesses multiple Gibbs measures. In this paper, we consider a three-state SOS (solid-on-solid) model on a Cayley tree. We reduce description of Gibbs measures to solving of a non-linear functional equation, each solution of which corresponds to a Gibbs measure.We give some sufficiency conditions on the existence of multiple Gibbs measures for the model. We give a review of some known (translation-invariant, periodic, non-periodic) Gibbs measures of the model and compare them with our new measures. We show that the Gibbs measures found in the paper differ from the known Gibbs measures, i.e, we show that these measures are new.

THERMODYNAMIC FORMALISM FOR AMENABLE GROUPS AND COUNTABLE STATE SPACES

Abstract Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties, such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that nontrivially satisfy the conditions for having this equivalence and a nonempty range of inverse temperatures where uniqueness holds.

On Gibbs measures and topological solitons of exterior equivariant wave maps

We consider k -equivariant wave maps from the exterior spatial domain \mathbb{R}^{3} \setminus B(0,1) into the target \mathbb{S}^{3} . This model has infinitely many topological solitons Q_{n,k} , which are indexed by their topological degree n\in \mathbb{Z} . For each n\in \mathbb{Z} and k\geq 1 , we prove the existence and invariance of a Gibbs measure supported on the homotopy class of Q_{n,k} . As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.

Gibbs Properties of the Bernoulli Field on Inhomogeneous Trees under the Removal of Isolated Sites

We consider the i.i.d. Bernoulli field p with occupation density p 2 (0; 1) on a possibly non-regular countably in finite tree with bounded degrees. For large p, we show that the quasilocal Gibbs property, i.e. compatibility with a suitable quasilocal speci fication, is lost under the deterministic transformation which removes all isolated ones and replaces them by zeros, while a quasilocal specifi cation does exist at small p. Our results provide an example for an independent field in a spatially nonhomogeneous setup which loses the quasilocal Gibbs property under a local deterministic transformation.