180 research outputs found
Cutoff for the noisy voter model
Given a continuous time Markov Chain on a finite set , the
associated noisy voter model is the continuous time Markov chain on
, which evolves in the following way: (1) for each two sites and
in , the state at site changes to the value of the state at site
at rate ; (2) each site rerandomizes its state at rate 1. We show that
if there is a uniform bound on the rates and the corresponding
stationary distributions are almost uniform, then the mixing time has a sharp
cutoff at time with a window of order 1. Lubetzky and Sly proved
cutoff with a window of order 1 for the stochastic Ising model on toroids; we
obtain the special case of their result for the cycle as a consequence of our
result. Finally, we consider the model on a star and demonstrate the surprising
phenomenon that the time it takes for the chain started at all ones to become
close in total variation to the chain started at all zeros is of smaller order
than the mixing time.Comment: Published at http://dx.doi.org/10.1214/15-AAP1108 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the existence and non-existence of finitary codings for a class of random fields
We study the existence of finitary codings (also called finitary homomorphisms or finitary factor maps) from a finite-valued i.i.d. process to certain random fields. For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the existence of the above coding: this yields a large class of Bernoulli shifts for which no such coding exists. Conversely, we show that for the stationary distribution of a monotone exponentially ergodic probabilistic cellular automaton such a coding does exist. The construction of the coding is partially inspired by the Propp-Wilson algorithm for exact simulation. In particular, combining our results with a theorem of Martinelli and Olivieri, we obtain the fact that for the plus state for the ferromagnetic Ising model on , , there is such a coding when the interaction parameter is below its critical value and there is no such coding when the interaction parameter is above its critical value
Percolation and the hard-core lattice gas model
AbstractRecently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks.First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least Pc(1 â Pc), where Pc is the critical probability for site percolation on that graph.Then, for the hard-core model on bi-partite graphs, we study the probability that a given vertex v is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a âpath of disagreementâ from v to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space.Finally, we discuss the hard-core model on Zd with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative
Quantisation of Conformal Fields in Three-dimensional Anti-de Sitter Black Hole Spacetime
Utilizing the conformal-flatness nature of 3-dim. Anti-de Sitter (AdS_3)
black hole solution of Banados, Teitelboim and Zanelli, the quantisation of
conformally-coupled scalar and spinor fields in this background spacetime is
explicitly carried out. In particular, mode expansion forms and propagators of
the fields are obtained in closed forms. The vacuum in this conformally-coupled
field theories in AdS_3 black hole spacetime, which is conformally-flat, is the
conformal vacuum which is unique and has global meaning. This point
particularly suggests that now the particle production by AdS_3 black hole
spacetime should be absent. General argument establishing the absence of real
particle creation by AdS_3 black hole spacetime for this case of conformal
triviality is provided. Then next, using the explicit mode expansion forms for
conformally-coupled scalar and spinor fields, the bosonic and fermionic
superradiances are examined and found to be absent confirming the expectation.Comment: 51 pages, Revtex, version to appear in Int. J. Mod. Phys.
Supergeometry of Three Dimensional Black Holes
We show how the supersymmetric properties of three dimensional black holes
can be obtained algebraically. The black hole solutions are constructed as
quotients of the supergroup by a discrete subgroup of its
isometry supergroup. The generators of the action of the isometry supergroup
which commute with these identifications are found. These yield the
supersymmetries for the black hole as found in recent studies as well as the
usual geometric isometries. It is also shown that in the limit of vanishing
cosmological constant, the black hole vacuum becomes a null orbifold, a
solution previously discussed in the context of string theory.Comment: 12 pages, harvmac, discussion of rotating black hole added, some
minor corrections, reference adde
Ultra--Planck Scattering in D=3 Gravity Theories
We obtain the high energy, small angle, 2-particle gravitational scattering
amplitudes in topologically massive gravity (TMG) and its two non-dynamical
constituents, Einstein and Chern--Simons gravity. We use 't Hooft's approach,
formally equivalent to a leading order eikonal approximation: one of the
particles is taken to scatter through the classical spacetime generated by the
other, which is idealized to be lightlike. The required geometries are derived
in all three models; in particular, we thereby provide the first explicit
asymptotically flat solution generated by a localized source in TMG. In
contrast to =4, the metrics are not uniquely specified, at least by naive
asymptotic requirements -- an indeterminacy mirrored in the scattering
amplitudes. The eikonal approach does provide a unique choice, however. We also
discuss the discontinuities that arise upon taking the limits, at the level of
the solutions, from TMG to its constituents, and compare with the analogous
topologically massive vector gauge field models.Comment: 20 pages, preprint BRX TH--337, DAMTP R93/5, ADP-93-204/M1
Back-reaction of a conformal field on a three-dimensional black hole
The first order corrections to the geometry of the (2+1)-dimensional black
hole due to back-reaction of a massless conformal scalar field are computed.
The renormalized stress energy tensor used as the source of Einstein equations
is computed with the Green function for the black-hole background with
transparent boundary conditions. This tensor has the same functional form as
the one found in the nonperturbative case which can be exactly solved. Thus, a
static, circularly symmetric and asymptotically anti-de Sitter black hole
solution of the semiclassical equations is found. The corrections to the
thermodynamic quantities are also computed.Comment: 12 pages, RevTeX, no figure
Curvature singularity of the distributional BTZ black hole geometry
For the non-rotating BTZ black hole, the distributional curvature tensor
field is found. It is shown to have singular parts proportional to a
-distribution with support at the origin. This singularity is related,
through Einstein field equations, to a point source. Coordinate invariance and
independence on the choice of differentiable structure of the results are
addressed.Comment: Latex, 7 page
Exact Results for the BTZ Black Hole
In this review, we summarize exact results for the three-dimensional BTZ
black hole. We use rigorous mathematical results to clarify the general
structure and properties of this black hole spacetime and its microscopic
description. In particular, we study the formation of the black hole by point
particle collisions, leading to an exact analytic determination of the Choptuik
scaling parameter. We also show that a `No Hair Theorem' follows immediately
from a mathematical theorem of hyperbolic geometry, due to Sullivan. A
microscopic understanding of the Bekenstein-Hawking entropy, and decay rate for
massless scalars, is shown to follow from standard results of conformal field
theory.Comment: 24 pages, Latex, Review article to appear in Int. J. Mod. Phys. D, v2
additional reference
The 2+1 Kepler Problem and Its Quantization
We study a system of two pointlike particles coupled to three dimensional
Einstein gravity. The reduced phase space can be considered as a deformed
version of the phase space of two special-relativistic point particles in the
centre of mass frame. When the system is quantized, we find some possibly
general effects of quantum gravity, such as a minimal distances and a foaminess
of the spacetime at the order of the Planck length. We also obtain a
quantization of geometry, which restricts the possible asymptotic geometries of
the universe.Comment: 59 pages, LaTeX2e, 9 eps figure
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