Sobre o modelo de percolação de grau restrito

Abstract

This thesis investigates the Constrained-Degree Percolation Model in a Random Environment (CDPRE) on the square lattice \( \mathbb{L}^2 \). In this model, each site \( v \) is assigned an independent random degree constraint \( \kappa_v \in \{0,1,2,3\} \), according to a probability distribution \( \rho \). Each edge \( e \) attempts to open at a random time \( U_e \sim U(0,1) \), and the opening is allowed only if, at that time, both of its endpoints have degrees strictly smaller than their respective constraints. We begin by reviewing known results regarding the structure of the model. Then, we present new contributions in the two-dimensional setting: we prove the almost sure uniqueness of the infinite open cluster in the supercritical phase, establish the positivity of the connectivity function and derive a Central Limit Theorem for the density of open edges. Furthermore, for higher dimensions and under deterministic constraints, we show that the probability of local events is differentiable with respect to time and we establish an adapted formulation of Russo's formula.