9,401 research outputs found
Stochastic domination for the Ising and fuzzy Potts models
We discuss various aspects concerning stochastic domination for the Ising
model and the fuzzy Potts model. We begin by considering the Ising model on the
homogeneous tree of degree , \Td. For given interaction parameters ,
and external field h_1\in\RR, we compute the smallest external field
such that the plus measure with parameters and dominates
the plus measure with parameters and for all .
Moreover, we discuss continuity of with respect to the three
parameters , , and also how the plus measures are stochastically
ordered in the interaction parameter for a fixed external field. Next, we
consider the fuzzy Potts model and prove that on \Zd the fuzzy Potts measures
dominate the same set of product measures while on \Td, for certain parameter
values, the free and minus fuzzy Potts measures dominate different product
measures. For the Ising model, Liggett and Steif proved that on \Zd the plus
measures dominate the same set of product measures while on \T^2 that
statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure
Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree
We continue our study of the full set of translation-invariant splitting
Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for
the -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 .
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 extremal TISGMs.
For the Cayley tree of order two we give explicit formulae and some numerical
values.Comment: 44 pages. To appear in Random Structures and Algorithm
Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts model with a Kac-type interaction
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.Comment: 20 page
Gibbs properties of the fuzzy Potts model on trees and in mean field
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 non-Gibbsianness happens
is given. The results for trees are somewhat less explicit, but we do show for
general trees that non-Gibbsianness of the fuzzy Potts model happens exactly
for those temperatures where the underlying Potts model has multiple Gibbs
measures
On Markov Chains with Uncertain Data
In this paper, a general method is described to determine uncertainty intervals for performance measures of Markov chains given an uncertainty region for the parameters of the Markov chains. We investigate the effects of uncertainties in the transition probabilities on the limiting distributions, on the state probabilities after n steps, on mean sojourn times in transient states, and on absorption probabilities for absorbing states. We show that the uncertainty effects can be calculated by solving linear programming problems in the case of interval uncertainty for the transition probabilities, and by second order cone optimization in the case of ellipsoidal uncertainty. Many examples are given, especially Markovian queueing examples, to illustrate the theory.Markov chain;Interval uncertainty;Ellipsoidal uncertainty;Linear Programming;Second Order Cone Optimization
On the prevalence of non-Gibbsian states in mathematical physics
Gibbs measures are the main object of study in equilibrium statistical
mechanics, and are used in many other contexts, including dynamical systems and
ergodic theory, and spatial statistics. However, in a large number of natural
instances one encounters measures that are not of Gibbsian form. We present
here a number of examples of such non-Gibbsian measures, and discuss some of
the underlying mathematical and physical issues to which they gave rise
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