Cutoff for the noisy voter model

Given a continuous time Markov Chain $\{q(x,y)\}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on $\{0,1\}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates $\{q(x,y)\}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $\log|S|/2$ 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 $bbbz^d$, $d geq 2$, 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 − 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 ‘path of disagreement’ 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 $OSp(1|\,2;R)$ 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 $D$=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 $\delta$-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