Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie--Weiss Ising model and includes as well all ferromagnetic Curie--Weiss Potts and Curie--Weiss Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that ``ferromagnetism'' is not however in itself sufficient and also study in some detail the Curie--Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie--Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a ``formula'' for the extension which is valid in many cases.Comment: Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coupling Surfaces and Weak Bernoulli in One and Higher Dimensions

AbstractWe propose a notion of weak Bernoulli in all dimensions which generalizes the usual definition in dimension 1. The key idea is the concept of acoupling surface. We relate this notion to previously studied properties and discuss a number of possible variants in dimension 1. We also show that the Ising model, at low temperature, is weak Bernoulli with an explicit description of the coupling surface

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

Time-Symmetric Initial Data for Multi-Body Solutions in Three Dimensions

Time-symmetric initial data for two-body solutions in three dimensional anti-deSitter gravity are found. The spatial geometry has constant negative curvature and is constructed as a quotient of two-dimensional hyperbolic space. Apparent horizons correspond to closed geodesics. In an open universe, it is shown that two black holes cannot exist separately, but are necessarily enclosed by a third horizon. In a closed universe, two separate black holes can exist provided there is an additional image mass.Comment: 12 pages, harvmac macro, minor changes in wordin

Lattice Universes in 2+1-dimensional gravity

Lattice universes are spatially closed space-times of spherical topology in the large, containing masses or black holes arranged in the symmetry of a regular polygon or polytope. Exact solutions for such spacetimes are found in 2+1 dimensions for Einstein gravity with a non-positive cosmological constant. By means of a mapping that preserves the essential nature of geodesics we establish analogies between the flat and the negative curvature cases. This map also allows treatment of point particles and black holes on a similar footing.Comment: 14 pages 7 figures, to appear in Festschrift for Vince Moncrief (CQG

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

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

Gott Time Machines, BTZ Black Hole Formation, and Choptuik Scaling

We study the formation of BTZ black holes by the collision of point particles. It is shown that the Gott time machine, originally constructed for the case of vanishing cosmological constant, provides a precise mechanism for black hole formation. As a result, one obtains an exact analytic understanding of the Choptuik scaling.Comment: 6 pages, Late

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

Thermodynamics and Evaporation of the 2+1-D Black Hole

The properties of canonical and microcanonical ensembles of a black hole with thermal radiation and the problem of black hole evaporation in 3-D are studied. In 3-D Einstein-anti-de Sitter gravity we have two relevant mass scales, $m_c=1/G$, and $m_p=(\hbar^2\Lambda/G)^{1/3}$, which are particularly relevant for the evaporation problem. It is argued that in the `weak coupling' regime $\Lambda<(\hbar G)^{-2}$, the end point of an evaporating black hole formed with an initial mass $m_0>m_p$, is likely to be a stable remnant in equilibrium with thermal radiation. The relevance of these results for the information problem and for the issue of back reaction is discussed. In the `strong coupling' regime, $\Lambda>(\hbar G)^{-2}$ a full fledged quantum gravity treatment is required. Since the total energy of thermal states in anti-de Sitter space with reflective boundary conditions at spatial infinity is bounded and conserved, the canonical and microcanonical ensembles are well defined. For a given temperature or energy black hole states are locally stable. In the weak coupling regime black hole states are more probable then pure radiation states.Comment: 11 pages, TAUP 2141/94, Late