arXiv subjects
Publications and source records attributed to Leonardo Rolla.
In this paper we describe the subcritical contact process on $\Z^d$ for large times, starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region.
We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $\mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.
In the present work we shall describe and apply the techniques of the Renormalization Group - based in data rescaling and operator renormalizing - and of Homogenization - that substitutes, in a certain limit, a periodically inhomogeneous medium by a homogeneous one - for the study of asymptotic behavior of solutions of PDE's. One class of problems studied in this dissertation consists of nonlinear, diffusive differential equations with periodic coefficients, that supposedly model the diffusion in a heterogeneous medium. Our study of these problems involves arguments originated from the two techniques described above. This makes, in certain sense, a connection between both of them. Another class of problems studied here consists of nonlinear, diffusive equations with time dependent coefficients, originated from montecarlo simulation of the asymptotic behavior of the averaged solutions of stochastic differential equations modeling the phenomenon of two-phase flow in porous media. Using the Renormalization Group we shall classify the problems in accordance with their qualitative behavior. This classification will be verified numerically. We shall also study the curve of phase transition that separates asymptotic regimes predominantly linear and nonlinear. The main results presented in this work are given by a numerical version of the Renormalization Group, which will be described in detail in the body of this dissertation.