Spectral properties of the renormalization group at infinite temperature

The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations.Comment: 12 page

Renormalization group maps for Ising models in lattice gas variables

Real space renormalization group maps, e.g., the majority rule transformation, map Ising type models to Ising type models on a coarser lattice. We show that each coefficient of the renormalized Hamiltonian in the lattice gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.Comment: 22 pages, 9 color postscript figures; minor revisions in version

Problems with the definition of renormalized Hamiltonians for momentum-space renormalization transformations

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.Comment: 14 pages, late

A quasilocal calculation of tidal heating

We present a method for computing the flux of energy through a closed surface containing a gravitating system. This method, which is based on the quasilocal formalism of Brown and York, is illustrated by two applications: a calculation of (i) the energy flux, via gravitational waves, through a surface near infinity and (ii) the tidal heating in the local asymptotic frame of a body interacting with an external tidal field. The second application represents the first use of the quasilocal formalism to study a non-stationary spacetime and shows how such methods can be used to study tidal effects in isolated gravitating systems.Comment: REVTex, 4 pages, 1 typo fixed, standard sign convention adopted for the Newtonian potential, a couple of lines added to the discussion of gauge dependent term

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on Zd\Z^d

In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity $\tau_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.Comment: 6 page

Cosmological Black Holes on Branes

We examined analytically a cosmological black hole domain wall system. Using the C-metric construction we derived the metric for the spacetime describing an infinitely thin domain wall intersecting a cosmological black hole. We studied the behaviour of the scalar field describing a self-interacting cosmological domain wall and find the approximated solution valid for large distances. The thin wall approximation and the back raection problem were elaborated finding that the topological kink solution smoothed out singular behaviour of the zero thickness wall using a core topological and hence thick domain wall. We also analyze the nucleation of cosmological black holes on and in the presence of a domain walls and conclude that the domain wall will nucleate small black holes on it rather than large ones inside.Comment: 13 pages, Revtex, to be published in Phys.Rev. D1

Expanding and Collapsing Scalar Field Thin Shell

This paper deals with the dynamics of scalar field thin shell in the Reissner-Nordstr$\ddot{o}$m geometry. The Israel junction conditions between Reissner-Nordstr$\ddot{o}$m spacetimes are derived, which lead to the equation of motion of scalar field shell and Klien-Gordon equation. These equations are solved numerically by taking scalar field model with the quadratic scalar potential. It is found that solution represents the expanding and collapsing scalar field shell. For the better understanding of this problem, we investigate the case of massless scalar field (by taking the scalar field potential zero). Also, we evaluate the scalar field potential when $p$ is an explicit function of $R$. We conclude that both massless as well as massive scalar field shell can expand to infinity at constant rate or collapse to zero size forming a curvature singularity or bounce under suitable conditions.Comment: 15 pages, 11 figure

High Temperature Expansions and Dynamical Systems

We develop a resummed high-temperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Perron-Frobenius operator of weakly coupled map lattices.Comment: 33 pages, Latex; [email protected]; [email protected]

On the absence of ferromagnetism in typical 2D ferromagnets

We consider the Ising systems in $d$ dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions which decay with a power, $s$, of the distance. The physical context of such models is discussed; primarily this is $d=2$ and $s=3$ where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above $d$ and does not exceed $d+1$, then for all temperatures, the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding $d+1$ (with $d\ge2$) magnetic order can occur.Comment: 15 pages, CMP style fil

Maximally incompressible neutron star matter

Relativistic kinetic theory, based on the Grad method of moments as developed by Israel and Stewart, is used to model viscous and thermal dissipation in neutron star matter and determine an upper limit on the maximum mass of neutron stars. In the context of kinetic theory, the equation of state must satisfy a set of constraints in order for the equilibrium states of the fluid to be thermodynamically stable and for perturbations from equilibrium to propagate causally via hyperbolic equations. Application of these constraints to neutron star matter restricts the stiffness of the most incompressible equation of state compatible with causality to be softer than the maximally incompressible equation of state that results from requiring the adiabatic sound speed to not exceed the speed of light. Using three equations of state based on experimental nucleon-nucleon scattering data and properties of light nuclei up to twice normal nuclear energy density, and the kinetic theory maximally incompressible equation of state at higher density, an upper limit on the maximum mass of neutron stars averaging 2.64 solar masses is derived.Comment: 8 pages, 2 figure