249 research outputs found
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
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
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
Concentration inequalities for random fields via coupling
We present a new and simple approach to concentration inequalities for
functions around their expectation with respect to non-product measures, i.e.,
for dependent random variables. Our method is based on coupling ideas and does
not use information inequalities. When one has a uniform control on the
coupling, this leads to exponential concentration inequalities. When such a
uniform control is no more possible, this leads to polynomial or
stretched-exponential concentration inequalities. Our abstract results apply to
Gibbs random fields, in particular to the low-temperature Ising model which is
a concrete example of non-uniformity of the coupling.Comment: New corrected version; 22 pages; 1 figure; New result added:
stretched-exponential inequalit
Partially ordered models
We provide a formal definition and study the basic properties of partially
ordered chains (POC). These systems were proposed to model textures in image
processing and to represent independence relations between random variables in
statistics (in the later case they are known as Bayesian networks). Our chains
are a generalization of probabilistic cellular automata (PCA) and their theory
has features intermediate between that of discrete-time processes and the
theory of statistical mechanical lattice fields. Its proper definition is based
on the notion of partially ordered specification (POS), in close analogy to the
theory of Gibbs measure. This paper contains two types of results. First, we
present the basic elements of the general theory of POCs: basic geometrical
issues, definition in terms of conditional probability kernels, extremal
decomposition, extremality and triviality, reconstruction starting from
single-site kernels, relations between POM and Gibbs fields. Second, we prove
three uniqueness criteria that correspond to the criteria known as bounded
uniformity, Dobrushin and disagreement percolation in the theory of Gibbs
measures.Comment: 54 pages, 11 figures, 6 simulations. Submited to Journal of Stat.
Phy
Percolation in the Harmonic Crystal and Voter Model in three dimensions
We investigate the site percolation transition in two strongly correlated
systems in three dimensions: the massless harmonic crystal and the voter model.
In the first case we start with a Gibbs measure for the potential,
, , and , a scalar height variable, and define
occupation variables for . The probability
of a site being occupied, is then a function of . In the voter model we
consider the stationary measure, in which each site is either occupied or
empty, with probability . In both cases the truncated pair correlation of
the occupation variables, , decays asymptotically like .
Using some novel Monte Carlo simulation methods and finite size scaling we find
accurate values of as well as the critical exponents for these systems.
The latter are different from that of independent percolation in , as
expected from the work of Weinrib and Halperin [WH] for the percolation
transition of systems with [A. Weinrib and B. Halperin,
Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent
is very close to the predicted value of 2 supporting the conjecture by WH
that is exact.Comment: 8 figures. new version significantly different from the old one,
includes new results, figures et
Conditional Intensity and Gibbsianness of Determinantal Point Processes
The Papangelou intensities of determinantal (or fermion) point processes are
investigated. These exhibit a monotonicity property expressing the repulsive
nature of the interaction, and satisfy a bound implying stochastic domination
by a Poisson point process. We also show that determinantal point processes
satisfy the so-called condition which is a general form of
Gibbsianness. Under a continuity assumption, the Gibbsian conditional
probabilities can be identified explicitly.Comment: revised and extende
Translation-invariance of two-dimensional Gibbsian point processes
The conservation of translation as a symmetry in two-dimensional systems with
interaction is a classical subject of statistical mechanics. Here we establish
such a result for Gibbsian particle systems with two-body interaction, where
the interesting cases of singular, hard-core and discontinuous interaction are
included. We start with the special case of pure hard core repulsion in order
to show how to treat hard cores in general.Comment: 44 pages, 6 figure
Chaotic temperature dependence at zero temperature
We present a class of examples of nearest-neighbour, boubded-spin models, in
which the low-temperature Gibbs measures do not converge as the temperature is
lowered to zero, in any dimension
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
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