Limiting Gibbs Measures Research Articles

Limiting Gibbs measures of the [formula omitted]-state Potts model with competing interactions

We consider q-state Potts model in the presence of competing two nearest interactions and prolonged next nearest interactions on a Cayley tree of order two. We derive the recursion relations, which determine the free energy of a q-state Potts model on the Cayley tree. We exactly solve the phase transitions problem for the model, namely we calculate the critical surface such that there is a phase transition above it, and a single Gibbs state found elsewhere. We study the periodicity of the phases (Gibbs measures) by means of Lyapunov exponents.

Limiting Gibbs Measures of Theq-State Potts Model with Competing Interactions Hasan Akin and Suleyman Ulusoy

Limiting Gibbs Measures of Theq-State Potts Model with Competing Interactions Hasan Akin and Suleyman Ulusoy

Translation‐invariant generalized P‐adic Gibbs measures for the Ising model on Cayley trees

In the present paper, we analyze the occurrence of a phase transition by the use of a p‐adic probability theory. To carry out this research, generalized p‐adic Gibbs measures are investigated, for the Ising model on the Cayley tree owing to the fact that this specific model has broad theoretical and practical applications. To explore translation‐invariant generalized p‐adic Gibbs measures, we describe the set of fixed points of the Ising‐Potts mapping which appears by means of renormalization group. This description allows us to establish the phase transition. In the real setting, the phase transition yields the singularity of the limiting Gibbs measures. However, we show that the generalized p‐adic Gibbs measures do not exhibit the mentioned type of singularity; this phenomena is called a strong phase transition.

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is $$2^{q}-1$$ . In this paper for each TISGM $$\mu $$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $$\mu $$ .

Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V_N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V_N) and the limiting two-overlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Generalized Random Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova (2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra identities hold exactly in the limit. As a corollary, the limiting array of overlaps is ultrametric and the limiting Gibbs measure has the same law as a Ruelle probability cascade.

On the overlap distribution of Branching Random Walks

In this paper, we study the overlap distribution and Gibbs measure of the Branching Random Walk with Gaussian increments on a binary tree. We first prove that the Branching Random Walk is 1 step Replica Symmetry Breaking and give a precise form for its overlap distribution, verifying a prediction of Derrida and Spohn. We then prove that the Gibbs measure of this system satisfies the Ghirlanda-Guerra identities. As a consequence, the limiting Gibbs measure has Poisson-Dirichlet statistics. The main technical result is a proof that the overlap distribution for the Branching Random Walk is supported on the set $\{0,1\}$.

Asymptotic domino statistics in the Aztec diamond

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

Random overlap structures: properties and applications to spin glasses

Random overlap structures (ROSt’s) are random elements on the space of probability measures on the unit ball of a Hilbert space, where two measures are identified if they differ by an isometry. In spin glasses, they arise as natural limits of Gibbs measures under the appropriate algebra of functions. We prove that the so called ‘cavity mapping’ on the space of ROSt’s is continuous, leading to a proof of the stochastic stability conjecture for the limiting Gibbs measures of a large class of spin glass models. Similar arguments yield the proofs of a number of other properties of ROSt’s that may be useful in future attempts at proving the ultrametricity conjecture. Lastly, assuming that the ultrametricity conjecture holds, the setup yields a constructive proof of the Parisi formula for the free energy of the Sherrington–Kirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr.

On phase transitions of the Potts model with three competing interactions on Cayley tree

In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12, No. 2, 141-156] on phase transition for the Ising model to the Potts model setting.

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T c,k =J/arctan (1/k) the limiting Gibbs measure is unique, and for T<T c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k}<k_{0}<k\) then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k 0≠k′0. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1}<T_{c,k}\) then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}<k_{0}<k}{\mathcal{G}}_{k,k_{0}})\).

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

A remark on the infinite-volume Gibbs measures of spin glasses

In this note, we point out that infinite-volume Gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a Hilbert space. This simple observation follows from a result of Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs measures can then be studied as single well-defined objects. This approach naturally extends the space of random overlap structures as defined by Aizenman et al. We discuss the Ruelle probability cascades and the stochastic stability within this framework. As an application, we use an idea of Parisi and Talagrand to prove that if a sequence of finite-volume Gibbs measures satisfies the Ghirlanda–Guerra identities, then the infinite-volume measure must be singular as a measure on a Hilbert space.

Limiting Gibbs measures of Potts model with countable set of spin values

We consider a nearest-neighbor Potts model, with countable spin values Φ = { 0 , 1 , … } , and nonzero external field, on a Cayley tree of order k (with k + 1 neighbors). We study translation-invariant ‘splitting’ Gibbs measures. The problem is reduced to the description of the solutions of some infinite system of equations. We give full description of the class of probabilistic measures ν on Φ such that our infinite system of equations has unique solution with respect to each element of this class. In particular we describe the Poisson measures which are Gibbsian.

On the three state Potts model with competing interactions on the Bethe lattice

In the present paper the three state Potts model with competing binary interactions (with couplingsJ andJp) on thesecond order Bethe lattice is considered. The recurrent equations for the partition functions are derived.For Jp = 0, by means of a construction of a special class of limiting Gibbs measures, it is shown howthese equations are related to the surface energy of the Hamiltonian. This relation reducesthe problem of describing the limit Gibbs measures to that of finding solutions of anonlinear functional equation. Moreover, the set of ground states of the one level model iscompletely described. Using this fact, we find Gibbs measures (pure phases) associatedwith the translation-invariant ground states. The critical temperature is found exactly andthe phase diagram is presented. The free energies corresponding to translation-invariantGibbs measures are found. Certain physical quantities are calculated as well.

The Potts model on Zd with countable set of spin values

The Potts model with countable set Φ of spin values on Zd is considered. It is proved that with respect to Poisson distribution on Φ the set of limiting Gibbs measures is not empty.

Group representation of the Cayley forest and some of its applications

Cayley forests and products of Cayley trees of order are represented as subgroups in the free product of cyclic groups () of order 2. The automorphism groups of these objects are determined. We give a complete description of the sets of translation-invariant and periodic Gibbs measures for the Ising model on a Cayley forest. We construct a new class of limiting Gibbs measures for the inhomogeneous Ising model on a Cayley tree. We find sufficient conditions for random walks in periodic random media on the forest to be never returning provided that the jumps of the walking particle are bounded.

Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous ising model

We prove the existence of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model on a Cayley tree and describe them constructively.