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Register a new userGradient Gibbs measures for the SOS model with countable values on a Cayley tree
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We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.
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We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.
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.
We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension $d=2$, while there are ``gradient Gibbs measures describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In $d=3$ where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.
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 $lambda > 0$. 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 $lambda>0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $lambda = 1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case, the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular, this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $lambda = 1$.
We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernandez, den Hollander and Mart{i}nez for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.
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