216 research outputs found
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]
Proof of a conjecture of N. Konno for the 1D contact process
Consider the one-dimensional contact process. About ten years ago, N. Konno
stated the conjecture that, for all positive integers , the upper
invariant measure has the following property: Conditioned on the event that
is infected, the events All sites are healthy and All
sites are healthy are negatively correlated. We prove (a stronger
version of) this conjecture, and explain that in some sense it is a dual
version of the planar case of one of our results in \citeBHK.Comment: Published at http://dx.doi.org/10.1214/074921706000000031 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Critical percolation of free product of groups
In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability of a free product of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups . For finite groups these
equations are polynomial and can be explicitly written down. The expected
subcritical cluster size of the free product is also found in terms of the
subcritical cluster sizes of the factors. In particular, we prove that
for the Cayley graph of the modular group (with the
standard generators) is , the unique root of the polynomial
in the interval .
In the case when groups can be "well approximated" by a sequence of
quotient groups, we show that the critical probabilities of the free product of
these approximations converge to the critical probability of
and the speed of convergence is exponential. Thus for residually finite groups,
for example, one can restrict oneself to the case when each free factor is
finite.
We show that the critical point, introduced by Schonmann,
of the free product is just the minimum of for the factors
Some conditional correlation inequalities for percolation and related processes
Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events ``there is an open path from s to a' and ``there is an open path from s to b' are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997
Handling Markov Chains with Membrane Computing
In this paper we approach the problem of computing the nâth power of
the transition matrix of an arbitrary Markov chain through membrane computing. The
proposed solution is described in a semiâuniform way in the framework of P systems with
external output. The amount of resources required in the construction is polynomial in
the number of states of the Markov chain and in the power. The time of execution is
linear in the power and is independent of the number of states involved in the Markov
chain.Ministerio de EducaciĂłn y Ciencia TIN2005-09345-C04-0
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice
In first-passage percolation on the integer lattice, the Shape Theorem
provides precise conditions for convergence of the set of sites reachable
within a given time from the origin, once rescaled, to a compact and convex
limiting shape. Here, we address convergence towards an asymptotic shape for
cone-like subgraphs of the lattice, where . In particular, we
identify the asymptotic shapes associated to these graphs as restrictions of
the asymptotic shape of the lattice. Apart from providing necessary and
sufficient conditions for - and almost sure convergence towards this
shape, we investigate also stronger notions such as complete convergence and
stability with respect to a dynamically evolving environment.Comment: 23 pages. Together with arXiv:1305.6260, this version replaces the
old. The main results have been strengthened and an earlier error in the
statement corrected. To appear in J. Theoret. Proba
Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton-Watson-like processes
This paper deals with the problem of perfect sampling from a Gibbs measure
with infinite range interactions. We present some sufficient conditions for the
extinction of processes which are like supermartingales when large values are
taken. This result has deep consequences on perfect simulation, showing that
local modifications on the interactions of a model do not affect simulability.
We also pose the question to optimize over a class of sequences of sets that
influence the sufficient condition for the perfect simulation of the Gibbs
measure. We completely solve this question both for the long range Ising models
and for the spin models with finite range interactions.Comment: 28 page
A combinatorial approach to jumping particles II: general boundary conditions
International audienceWe consider a model of particles jumping on a row, the TASEP. From the point of view of combinatorics a remarkable feauture of this Markov chain is that Catalan numbers are involved in several entries of its stationary distribution. In a companion paper, we gave a combinatorial interpretaion and a simple proof of these observations in the simplest case where the particles enter, jump and exit at the same rate. In this paper we show how to deal with general rates
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