In this paper, we consider a Hard-Core $(HC)$ model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.
We consider the ferromagnetic n.n Ising model on Cayley trees in absence of external fields submitted to a modified majority rule transformation with overlapping cells already known to lead to non-Gibbsian measures. We describe the renormalized measures within the Generalized Gibbs framework and prove that they are Almost Gibbs at any temperature.
In this paper, we consider the Potts-SOS model where the spin takes values in the set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions. Moreover, we investigate whether these Gibbs measures are extremal or non-extremal in the set of all Gibbs measures.
In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology etc.
In this paper, we consider Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree. For this model we define Markov random fields with memory of length 2. By using a new approach, we obtain new sets of Gibbs measures of Ising-Vannimenus model on Cayley tree of order 2. We construct the recurrence equations corresponding Ising-Vannimenus model. We prove the Kolmogorov consistency condition. We investigate the translation-invariant and periodic non transition-invariant Gibbs measures for the model. We find new sets of Gibbs measures di…
In this paper, we consider the Ising-Vanniminus model on an arbitrary order Cayley tree. We generalize the results conjectured in [Chinese Journal of Physics, 54 (4), 635-649 (2016)] and [International Journal of Modern Physics, arXiv:1608.06178] for an arbitrary order Cayley tree. We establish existence and a full classification of translation invariant Gibbs measures with memory of length 2 associated with the model on arbitrary order Cayley tree. We construct the recurrence equations corresponding generalized ANNNI model. We satisfy the Kolmogorov \emph{consistency} condition. We propose a …
Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a non-linear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measu…
In this paper we investigate generalized Gibbs measure (GGM) for $p$-adic Hard-Core(HC) model with a countable set of spin values on a Cayley tree of order $k\geq 2$. This model is defined by $p$-adic parameters $λ_i$, $i\in \mathbb N$. We analyze $p$-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by $p$-adic version of Kolmogorov's theorem. We define $p$-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distribu…
Gibbs measure for the HC-Blume-Capel model in the case of a "wand" type graph on a Cayley tree
Namangan State University（纳曼干国立大学）；Institute of Mathematics named after V.I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan（乌兹别克斯坦共和国科学院V.I.罗曼诺夫斯基数学研究所）；Andijan State University（安集延国立大学）
We consider nearest-neighbor fertile hard-core models, with three states, on a homogeneous Cayley tree. It is known that there are four type of such models. We investigate all of them and describe translation-invariant and periodic hard-core Gibbs measures. Also we construct a continuum set of non-periodic Gibbs measures.
We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq 2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For $m=2$, in the anti-ferromagnetic (AFM) case, a symmetric TISGM is unique for all temperatures. In the ferromagnetic (FM) case, for $m=2$, the number of symmetric TISGMs varies with the temperature: here we identify a critical inverse temperature, $β^1_{\rm{cr}}$ ($=T_{\rm{cr}}^{\rm{STISG}}$) $\in (0,\infty)$ such that $\forall$ $0\leq β\leqβ^1_{\rm{cr}}$, there exists a uniqu…
In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12,EsRo10,BoEsRo13,JaKuBo14,Bo17]. The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value $θ_{\rm c}$ such that for $θ\le θ_{\rm c}$ there is a unique translation-invariant splitting Gibbs measure. For $θ_{\rm c}<θ$ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an asso…
In the paper the Ising model with competing $J_1$ and $J_2$ interactions with spin values $\pm 1$, on a Cayley tree of order 2 (with 3 neighbors) is considered . We study the structure of the ground states and verify the Peierls condition for the model. Our second result gives description of Gibbs measures for ferromagnetic Ising model with $J_1<0$ and $J_2=0$, using a contour argument which we also develop in the paper. By the argument we also study Gibbs measures for a natural generalization of the Ising model. We discuss some open problems and state several conjectures.\
In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $ρ\in\bq_p$. Such a measure is called {\it generalized $p$-adic quasi Gibbs measure}. When $ρ$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $ρ=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|ρ|_p$. Namely, in the first regime, one takes $ρ=\exp_p(J)$ for some $…
The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invari…
Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree
The United Arab Emirates University（阿联酋大学）；V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；Oriental University（东方大学）
We consider a nearest-neighbor four state hard-core (HC) model on the homogeneous Cayley tree of order $k$. The Hamiltonian of the model is considered on a set of "admissible" configurations. Admissibility is specified through a graph with four vertices. We first exhibit conditions (on the graph and on the parameters) under which the model has a unique Gibbs measure. Next we turn to some specific cases. Namely, first we study, in the case of a particular graph (the diamond), translation-invariant and periodic Gibbs measures. We provide in both cases the equations of the transition lines separa…
We consider both Hard-Core and Soft-Core Widom-Rowlinson models with spin values $-1,0,1$ on a Cayley tree of order $k\geq 2$ and we are interested in the Gibbs measures of the models. The models depend on 3 parameters: the order $k$ of the tree, $θ$ describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and $λ$ describing the intensity for particles. The Hard-Core Widom-Rowlinson model corresponds to the case $θ=0$. For the binary tree $k=2$, and for $k=3$ we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invari…
Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e. hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise e.g. in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interaction…
In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian …
We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order $d$ (which has $d + 1$ nearest neighbours), depending on repulsion strength $β$ between particles of different signs and on an activity parameter $λ$ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength $β$. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activit…
We consider the Widom-Rowlinson model on the lattice $\mathbb{Z}^d$ in two versions, comparing the cases of a hard-core repulsion and of a soft-core repulsion between particles carrying opposite signs. For both versions we investigate their dynamical Gibbs-non-Gibbs transitions under an independent stochastic symmetric spin-flip dynamics. While both models have a similar phase transition in the high-intensity regime in equilibrium, we show that they behave differently under time-evolution: The time-evolved soft-core model is Gibbs for small times and loses the Gibbs property for large enough t…
In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $ł$-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter $\r\in\bq_p$, for the $ł$-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed p…
Coupling from the past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes, which is a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned that spheres do not overlap with each other. We propose an alternative importance sampling based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases …
We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $λ> 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $λ>0$. Our result shows that…
It is well known that for a regular stable potential of pair interaction and a small value of activity one can define the corresponding Gibbs field (a measure on the space of configurations of points in $\mathbb{R}^d$). In this paper we consider a converse problem. Namely, we show that for a sufficiently small constant $\overlineρ_1$ and a sufficiently small function $\overlineρ_2(x)$, $x \in \mathbb{R}^d$, that is equal to zero in a neighborhood of the origin, there exist a hard core pair potential, and a value of activity, such that $\overlineρ_1$ is the density and $\overlineρ_2$ is the pai…
The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds impr…
We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factor…
We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $ł$, where the weight of an independent set $I$ is $ł^{|I|}$. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree $d$ and $ł<ł_c(d)$, where $ł_c(d)$ is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the $d$-regular infinite tree. We show that for $d \geq 3$, $ł$ just above $ł_c(…
Disordered systems such as spin glasses have been used extensively as models for high-dimensional random landscapes and studied from the perspective of optimization algorithms. In a recent paper by L. Addario-Berry and the second author, the continuous random energy model (CREM) was proposed as a simple toy model to study the efficiency of such algorithms. The following question was raised in that paper: what is the threshold $β_G$, at which sampling (approximately) from the Gibbs measure at inverse temperature $β$ becomes algorithmically hard? This paper is a first step towards answering this…
A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs asymptotically until just below its condensation threshold, showing that it converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tr…
