Results 1  10
of
24
Gibbs properties of the fuzzy Potts model on trees and in mean field
, 2008
"... We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which nonGibbsianness happens is given. The results for trees are somewhat less explicit, but we do ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which nonGibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general trees that nonGibbsianness of the fuzzy Potts model happens exactly for those temperatures where the underlying Potts model has multiple Gibbs measures. 1
The renormalizationgroup peculiarities of Griffiths and Pearce: What have we learned?,
 in Mathematical Results in Statistical Mechanics (Proceedings of the colloquium with the same name, MarseilleLuminy,
, 1998
"... ..."
Is the fuzzy Potts model Gibbsian?
, 2002
"... The fuzzy Potts model is obtained by looking at the Potts model with a pair of glasses that prevents distinguishing between some of the spin values. We show that the fuzzy Potts model on Z (d 2) is Gibbsian at high temeratures and nonGibbsian at low temperatures. ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The fuzzy Potts model is obtained by looking at the Potts model with a pair of glasses that prevents distinguishing between some of the spin values. We show that the fuzzy Potts model on Z (d 2) is Gibbsian at high temeratures and nonGibbsian at low temperatures.
Variational Principle and Almost Quasilocality for Renormalized Measures.
"... We restore part of the thermodynamic formalism for some renormalized measures that are known to be nonGibbsian. We first point out that a recent theory due to Pfister implies that for blocktransformed measures free energy and relative entropy densities exist and are conjugated convex functiona ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
We restore part of the thermodynamic formalism for some renormalized measures that are known to be nonGibbsian. We first point out that a recent theory due to Pfister implies that for blocktransformed measures free energy and relative entropy densities exist and are conjugated convex functionals. We then determine a necessary and sufficient condition for consistency with a specification that is quasilocal in a fixed direction. As corollaries we obtain consistency results for models with FKG monotonicity and for models with appropriate "continuity rates". For (noisy) decimations or projections of the Ising model these results imply almost quasilocality of the decimated "+" and "\Gamma" measures.
Majority rule at low temperatures on the square and triangular lattices
 J. Stat. Phys
, 1997
"... We consider the majority rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of [17], who proved the renormalized ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
We consider the majority rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of [17], who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero temperature majority rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.
Positive correlations in the fuzzy Potts model
 Ann. Appl. Probab
, 1998
"... The fuzzy Potts model arises by taking the qstate Potts model, and then identifying r of the Potts spins with the fuzzy spin 0, and the remaining q \Gamma r Potts spins with the fuzzy spin 1. Here we extend a result of L. Chayes by showing that the fuzzy Potts model has positive correlations. We a ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
The fuzzy Potts model arises by taking the qstate Potts model, and then identifying r of the Potts spins with the fuzzy spin 0, and the remaining q \Gamma r Potts spins with the fuzzy spin 1. Here we extend a result of L. Chayes by showing that the fuzzy Potts model has positive correlations. We also give an application to the percolationtheoretic behavior of the Potts model on Z 2 . 1 Introduction The qstate Potts model on a finite graph G = (V; E) is a random assignment of f1; : : : ; qgvalued spins to the vertices of G. The Gibbs measure ß G q;fi for the qstate Potts model on G at inverse temperature fi 0, is the measure on f1; : : : ; qg V which to each ¸ 2 f1; : : : ; qg V assigns probability ß G q;fi (¸) = 1 Z G q;fi exp 0 @ 2fi X hx;yi2E I f¸(x)=¸(y)g 1 A : (1) Here hx; yi denotes the edge connecting x; y 2 V , I A is the indicator function of the event A, and Z G q;fi is a normalizing constant making ß G q;fi a probability measure. This model is much...
Coloring Percolation Clusters At Random
, 2000
"... We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation on G, and then assigning a random color, chosen according to some prescribed probability distribution on the finite set f0; : : : ; r \Gamma 1g, to each of the connected components, ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation on G, and then assigning a random color, chosen according to some prescribed probability distribution on the finite set f0; : : : ; r \Gamma 1g, to each of the connected components, independently for different components. We call this the divide and color model, and study its percolation and Gibbs (quasilocality) properties, with emphasis on the case G = Z d . These properties turn out to depend heavily on the parameters of the model. For r = 2, an FKG inequality is also obtained. 1 Introduction The purpose of this paper is to introduce and study a simple and natural model for dependent colorings of the vertices of a (finite or infinite) graph G with vertex set V and edge set E. We allow r 2 different colors, denoted 0; 1; : : : ; r \Gamma 1. Besides r, the model has the additional parameters p and a 0 ; a 1 ; : : : ; a r\Gamma1 , all taking values in [0; 1], and s...