285 research outputs found
Phase transition and percolation in Gibbsian particle models
We discuss the interrelation between phase transitions in interacting lattice
or continuum models, and the existence of infinite clusters in suitable
random-graph models. In particular, we describe a random-geometric approach to
the phase transition in the continuum Ising model of two species of particles
with soft or hard interspecies repulsion. We comment also on the related
area-interaction process and on perfect simulation.Comment: Survey article, 25 page
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
R-local Delaunay inhibition model
Let us consider the local specification system of Gibbs point process with
inhib ition pairwise interaction acting on some Delaunay subgraph specifically
not con taining the edges of Delaunay triangles with circumscribed circle of
radius grea ter than some fixed positive real value . Even if we think that
there exists at least a stationary Gibbs state associated to such system, we do
not know yet how to prove it mainly due to some uncontrolled "negative"
contribution in the expression of the local energy needed to insert any number
of points in some large enough empty region of the space. This is solved by
introducing some subgraph, called the -local Delaunay graph, which is a
slight but tailored modification of the previous one. This kind of model does
not inherit the local stability property but satisfies s ome new extension
called -local stability. This weakened property combined with the local
property provides the existence o f Gibbs state.Comment: soumis \`{a} Journal of Statistical Physics 27 page
Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation
This paper presents three results on dependent site percolation on the square
lattice. First, there exists no positively associated probability measure on
{0,1}^{Z^2} with the following properties: a) a single infinite 0cluster exists
almost surely, b) at most one infinite 1*cluster exists almost surely, c) some
probabilities regarding 1*clusters are bounded away from zero. Second, we show
that coexistence of an infinite 1*cluster and an infinite 0cluster is almost
surely impossible when the underlying probability measure is ergodic with
respect to translations, positively associated, and satisfies the finite energy
condition. The third result analyses the typical structure of infinite clusters
of both types in the absence of positive association. Namely, under a slightly
sharpened finite energy condition, the existence of infinitely many disjoint
infinite self-avoiding 1*paths follows from the existence of an infinite
1*cluster. The same holds with respect to 0paths and 0clusters.Comment: 17 pages, 1 figur
Bell's Jump Process in Discrete Time
The jump process introduced by J. S. Bell in 1986, for defining a quantum
field theory without observers, presupposes that space is discrete whereas time
is continuous. In this letter, our interest is to find an analogous process in
discrete time. We argue that a genuine analog does not exist, but provide
examples of processes in discrete time that could be used as a replacement.Comment: 7 pages LaTeX, no figure
There are No Nice Interfaces in 2+1 Dimensional SOS-Models in Random Media
We prove that in dimension translation covariant Gibbs states
describing rigid interfaces in a disordered solid-on-solid (SOS) cannot exist
for any value of the temperature, in contrast to the situation in .
The prove relies on an adaptation of a theorem of Aizenman and Wehr.
Keywords: Disordered systems, interfaces, SOS-modelComment: 8 pages, gz-compressed Postscrip
Phase Transition in Ferromagnetic Ising Models with Non-Uniform External Magnetic Fields
In this article we study the phase transition phenomenon for the Ising model
under the action of a non-uniform external magnetic field. We show that the
Ising model on the hypercubic lattice with a summable magnetic field has a
first-order phase transition and, for any positive (resp. negative) and bounded
magnetic field, the model does not present the phase transition phenomenon
whenever , where is the external
magnetic field.Comment: 11 pages. Published in Journal of Statistical Physics - 201
Topological Signature of First Order Phase Transitions
We show that the presence and the location of first order phase transitions
in a thermodynamic system can be deduced by the study of the topology of the
potential energy function, V(q), without introducing any thermodynamic measure.
In particular, we present the thermodynamics of an analytically solvable
mean-field model with a k-body interaction which -depending on the value of k-
displays no transition (k=1), second order (k=2) or first order (k>2) phase
transition. This rich behavior is quantitatively retrieved by the investigation
of a topological invariant, the Euler characteristic, of some submanifolds of
the configuration space. Finally, we conjecture a direct link between the Euler
characteristic and the thermodynamic entropy.Comment: 6 pages, 2 figure
First-order transitions for some generalized XY models
In this note we demonstrate the occurrence of first-order transitions in
temperature for some recently introduced generalized XY models, and also point
out the connection between them and annealed site-diluted (lattice-gas)
continuous-spin models
Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles
In this paper we study a continuum version of the Potts model. Particles are
points in R^d, with a spin which may take S possible values, S being at least
3. Particles with different spins repel each other via a Kac pair potential. In
mean field, for any inverse temperature there is a value of the chemical
potential at which S+1 distinct phases coexist. For each mean field pure phase,
we introduce a restricted ensemble which is defined so that the empirical
particles densities are close to the mean field values. Then, in the spirit of
the Dobrushin Shlosman theory, we get uniqueness and exponential decay of
correlations when the range of the interaction is large enough. In a second
paper, we will use such a result to implement the Pirogov-Sinai scheme proving
coexistence of S+1 extremal DLR measures.Comment: 72 pages, 1 figur
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