Two-Dimensional Scaling Limits via Marked Nonsimple Loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE(6) and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure

Scaling limit for a drainage network model

We consider the two dimensional version of a drainage network model introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed by Fontes, Isopi, Newman and Ravishankar.Comment: 15 page

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\omega_n$ at time n converges exponentially fast to a final configuration $\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $\beta$, $\eta$, $\nu$ and $\gamma$, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ${\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\mathbb Z}^2$), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents. This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement.Comment: 15 page

Large deviations principle for Curie-Weiss models with random fields

In this article we consider an extension of the classical Curie-Weiss model in which the global and deterministic external magnetic field is replaced by local and random external fields which interact with each spin of the system. We prove a Large Deviations Principle for the so-called {\it magnetization per spin} $S_n/n$ with respect to the associated Gibbs measure, where $S_n/n$ is the scaled partial sum of spins. In particular, we obtain an explicit expression for the LDP rate function, which enables an extensive study of the phase diagram in some examples. It is worth mentioning that the model considered in this article covers, in particular, both the case of i.\,i.\,d.\ random external fields (also known under the name of random field Curie-Weiss models) and the case of dependent random external fields generated by e.\,g.\ Markov chains or dynamical systems.Comment: 11 page

Variabilidade espacial da textura de dois solos do Deserto Salino, no Estado do Rio Grande do Norte.

Este trabalho teve como objetivo avaliar a variabilidade espacial da textura do solo, em duas áreas do Deserto Salino, na região da planície aluvial do Rio Apodi/Mossoró-RN. Para atingir esse objetivo, utilizaram-se técnicas de geoestatística. As áreas foram georeferenciadas em imagens de satélite (LandSat TM-7), empregandose um GPS portátil. O esquema de amostragem foi ao acaso com 60 unidades amostrais, com volume de 1 dm3, coletadas na camada de 0 a 5 cm. Com exceção do teor de areia grossa no Vertissolo hidromórfico, todos os demais componentes granulométricos, em ambos os solos, apresentaram dependência espacial, sendo o esférico e o exponencial os modelos ajustados aos semivariogramas. O valor do efeito ?pepita? teve pequena contribuição na variância total dos dados e as variáveis apresentaram forte dependência espacial. Houve grande variação na distância até onde as variáveis estudadas apresentaram dependência espacial (alcance), tanto dentro das áreas como entre as áreas estudadas para a maioria das variáveis, indicando que a utilização de um mesmo esquema de amostragem não é apropriado quando se trabalha com muitas variáveis. Por meio dos mapas de ?krigagem?, identificaram-se padrões semelhantes de distribuição espacial para algumas características físicas em função da dinâmica da água

Analytical results for a Bessel function times Legendre polynomials class integrals

When treating problems of vector diffraction in electromagnetic theory, the evaluation of the integral involving Bessel and associated Legendre functions is necessary. Here we present the analytical result for this integral that will make unnecessary numerical quadrature techniques or localized approximations. The solution is presented using the properties of the Bessel and associated Legendre functions.Comment: 4 page

Desenvolvimento e precocidade de produção de coqueiros híbridos, consorciados com Gliricidia sepium.

ODS 2