Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.     

Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0. (3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. Received: 26 April 2001 / Accepted: 10 October 2001     

A Constructive Description of Ground States and Gibbs Measures for Ising Model with Two-Step Interactions on Cayley Tree

We consider the Ising model with (competing) two-step interactions and spin values ± 1, on a Cayley tree of order k ≥ 1. We constructively describe ground states and verify the Peierls condition for the model. We define notion of a contour for the model on the Cayley tree. Using a contour argument we show the existence of two different Gibbs measures.     

Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree

For the \(q\) -state Potts model on a Cayley tree of order \(k\ge 2\) it is well-known that at sufficiently low temperatures there are at least \(q+1\) translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is \(2^{q}-1\) . We prove that there are \([q/2]\) (where \([a]\) is the integer part of \(a\) ) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.     

Gibbs Distributions for Random Partitions Generated by a Fragmentation Process

In this paper we study random partitions of {1,…,n} where every cluster of size j can be in any of w _j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w _j , the time-reversed process is the discrete Marcus–Lushnikov coalescent process with affine collision rate K _i,j = a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton–Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions. Research supported in part by N.S.F. Grant DMS-0405779.     

A link between quantum and classical Potts models

We study ground states of quantum Potts models. We construct ground states of certaind-dimensional quantum models as Gibbs measures of ad-dimensional classical spin system. Our results imply that various phenomena of classical spin systems can also be found in quantum ground states.     

A Three State Hard-Core Model on a Cayley Tree

Abstract

We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ? λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure ^*. We also study periodic splitting Gibbs measures and show that the above model admits only translation - invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding von Neumann Algebras

In the present paper a model with competing ternary (J ₂) and binary (J ₁) interactions with spin values ±1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J ₁,J ₂ (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation-invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the unordered phase is a factor of type III _λ , where λ=exp{?2βJ ₂}, β>0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III _δ , otherwise they have type III₁.     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J₁) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.     

On Free Energies of the Ising Model on the Cayley Tree

We present, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). They include translation-invariant, periodic, Dobrushin-like b.c., as well as those corresponding to (recently discovered) weakly periodic Gibbs states. The weakly periodic measures are defined through a partition of the Cayley tree that induces a 4-edge-coloring on that tree. We compute the density of each color. We use these densities for computations of free energies corresponding to a weakly periodic b.c.     

High-spin states and signature inversion in odd-odd182Au

High-spin states in¹⁸²Au have been produced and studied via the¹⁵²Sm(³⁵Cl,5nγ)¹⁸²Au reaction. The level scheme consisting of the πh _9/2⊗νi _13/2 and πi _13/2⊗νi _13/2 bands has been established for the first time. The low spin signature inversion in both bands has been found. The observed signature inversion phenomena can be interpreted qualitatively using the pairing and deformation self-consistent cranked Wood-Saxon calculations.     

Influence of quenched nonmagnetic impurities on phase transitions in a three-dimensional diluted Potts model with spin state number q = 4

The influence of quenched nonmagnetic impurities on phase transitions and critical phenomena in the 3D Potts model with the spin state number q = 4 is studied using the Monte Carlo method. Systems with the linear size L = 20–32 and spin concentrations p = 1.00, 0.90, 0.65 are considered. The fourth order Binder cumulant method is used to demonstrate that in the strongly diluted regime, a phase transition of the second kind is observed in this model for the spin concentration p = 0.65, and a phase transition of the first kind is observed for the pure (p = 1.00) and weakly diluted (p = 0.90) models. The theory of finite-dimensional scaling is used to calculate the static critical parameters of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν.     

Condensation in Nongeneric Trees

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability m<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−m)N where N is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2β−2 if the weight w _n of a vertex of degree n is asymptotic to n ^−β .     

On the Zero-Temperature Limit of Gibbs States

We exhibit Lipschitz (and hence Hölder) potentials on the full shift ${\{0,1\}^{\mathbb{N}}}We exhibit Lipschitz (and hence H?lder) potentials on the full shift {0,1}^\mathbbN{\{0,1\}^{\mathbb{N}}} such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are “exponentially decaying” interactions on the configuration space {0,1}^{\mathbb Z}{\{0,1\}^{\mathbb Z}} for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space {0,1}^\mathbbZd{\{0,1\}^{\mathbb{Z}^{d}}}, d ≥ 3, we show that this non-convergence behavior can occur for the equilibrium states of finite-range interactions, that is, for locally constant potentials.     

Observation of a πh9/2 ⊗ νi13/2 oblate band in 188Tl

Excited states in ¹⁸⁸Tl have been studied experimentally using the ¹⁵⁷Gd (³⁵Cl, 4n) reaction at a beam energy of 170MeV. A rotational band built on the πh _9/2 ⊗ νi _13/2 configuration with oblate deformation has been established for ¹⁸⁸Tl. Based on the structure systematics of the oblate πh _9/2 ⊗ νi _13/2 bands in the heavier odd-odd Tl nuclei, we have tentatively proposed spin values for the new band in ¹⁸⁸Tl. The πh _9/2 ⊗ νi _13/2 oblate band in ¹⁸⁸Tl shows low-spin signature inversion, and it can be interpreted qualitatively by the two-quasiparticle plus rotor model including a J-dependent p-n residual interaction.     

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order $k\geqslant 1$ . It is known that the ‘splitting Gibbs measures’ of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geqslant 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.     

Phase transition properties of three-dimensional systems described by diluted potts model

Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.