Gastão A. Braga

Approximate similarity solutions to the Boussinesq equation

We derive a new cubic polynomial approximation to the similarity solution of the Boussinesq equation for one-dimensional, unconfined groundwater flow in a horizontal, initially dry aquifer with a power law inlet condition. This derivation builds upon the work of Lockington et al. (Adv. Wat. Resour. 23 (7) (2000) 725), where a quadratic approximation is constructed, and uses an additional condition for the determination of the (approximate) scaling function. The approximate analytical solutions are compared with the direct numerical solution obtained by Shampines method (ZAMM 53 (1973) 421). In all the tests performed, the new cubic approximation gives better results than the quadratic one, while it is only slightly harder to use.

Analyticity properties and a convergent expansion for the glueball mass and dispersion curve of strongly coupled Euclidean 2+1 lattice gauge theories

For certain strongly coupled (2/g2≡β>0 and small) Euclidean Z3 lattice gauge theories we show that the glueball mass m(β) associated with the truncated plaquette–plaquette correlation function admits the representation m(β)=−4 ln β+r(β). r(β)=∑∞n=0bnβn is a gauge group representation dependent function, analytic at β=0. A finite algorithm is given for determining bn. bn depends on a finite number of the β=0 Taylor series coefficients of the finite lattice correlation function at a finite number of points, increasing with n, of Z3.

Upper bounds on the critical temperature for the two-dimensional Blume-Emery-Griffiths model

We obtain rigorous upper bounds for the critical temperature associated with second-order phase transitions of the two-dimensional spin-1 BEG model for real values ofK andD coupling constants and forJ≥0. We use some correlation equalities and inequalities to show the exponential decay of the two-point function characterizing the disordered phase.

Analyticity of the d-Dimensional Bond Percolation Probability Around p=1

Let θ(p) be the percolation probability of a d-dimensional bond percolation process on Zd. We prove that 1−θ(p) can be written as an absolutely convergent series in powers of (1−p)/p, provided that |(1−p)/p| is sufficiently small. This implies that θ(p) is an analytic function of the complex variable p, around p=1.

Renormalization Group Analysis of Nonlinear Diffusion Equations with Periodic Coefficients

In this paper we present an efficient numerical approach based on the renormalization group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the renormalization group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on ℤd

Abstract In this short note we consider mixed short-long range independent bond percolation models on ℤk+d. Let puv 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 τxy is governed by the probability pxy. The result holds up to the critical point.

The Blume–Emery–Griffiths model at an infinitely many ground states interface and exponential decay of correlations at all non-zero temperatures

In this paper we study the spin–spin correlation function decay properties of the Blume–Emery–Griffiths (BEG) model with Hamiltonian located on the interface between the disordered and the anti-quadrupolar phases. On this interface, the BEG model has infinitely many ground state configurations. We show that, for any dimension d, there exists a parameter value, yd, below which the spin–spin correlation function with zero boundary condition decays exponentially fast at all non-zero temperatures. This result suggests that reentrant behaviour predicted by mean-field and numerical calculations may be absent for those values of parameters.

LOW TEMPERATURE PROPERTIES OF THE BLUME–EMERY–GRIFFITHS (BEG) MODEL IN THE REGION WITH AN INFINITE NUMBER OF GROUND STATE CONFIGURATIONS

For the low temperature Blume–Emery–Griffiths Zd, d ≥2, lattice model taking site spin values 0, +1, -1 we construct, using a polymer expansion, two pure states in the parameter region where there are an infinite number of configurations with minimal energy. Each state is invariant under translation by two lattice spacings and the two states are related by a unit translation. Using analyticity techniques we show that the truncated n-point function decays exponentially with an n-independent lower bound on the decay rate. For the truncated two-point function, we find the exact exponential decay rate in the limit β→∞.

EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS VIA THE GENERATING FUNCTION: A DIRECT METHOD

We consider statistical mechanics lattice models where the external field dependent partition function can be represented as a standard polymer system. Using this polymer representation and elementary complex analytic arguments, we obtain upper bounds and give a simple proof on the uniform (in n) exponential decay of the n-point truncated correlation function. We illustrate the method by applying it to the high and low temperature Ising model and to contour models.

A multiscale asymptotic analysis of time evolution equations on the complex plane

Using an appropriate norm on the space of entire functions, we extend to the complex plane the renormalization group method as developed by Bricmont et al. The method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems. The time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical N-vector model. We show that, for initial conditions belonging to a certain Frechet space of entire functions of exponential type, the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions.