RUI: First-passage Percolation and Other Disordered Systems

Cuny Baruch College, New York NY

Investigators

Abstract

Two models of disordered systems, first-passage percolation (FPP) and stochastic Ising dynamics with random initial state, are the main focus of this proposal. Rigorous unconditional results about the geometric features of lattice-based FPP models are hard to come by, largely due to technical difficulties associated with the anisotropy of the underlying lattice. Models of Euclidean FPP take place on graphs constructed from a homogeneous Poisson process and enjoy complete statistical invariance under all rigid motions. In this sense, Euclidean FPP is a natural setting in which to study geometric properties of FPP. One objective of this proposal is to refine, for Euclidean FPP, existing estimates of key fluctuation exponents associated with these models with the ultimate goal of proving interesting qualitative results. Another goal is to exploit a certain re-scaling relation that holds for Euclidean FPP to obtain results about the stochastic monotonicity of passage time viewed as a function of distance. Other closely related open questions under investigation concern issues such as how massively a local perturbation of the Poisson particle configuration affects minimizing paths and how far "parallel" geodesics typically travel before coalescence. In Stochastic Ising models, rules of dynamics interact with a random initial state to produce a system that evolves with time. The state at any particular time is a configuration of +/-1 spins associated with the sites of some regular lattice; the configuration at time 0 makes the spins i.i.d. random variables. For the homogeneous ferromagnetic models of this proposal, the dynamics tend to produce increasing agreement of spins for neighboring sites as time elapses. Key matters of investigation concern how quickly and in what sense the system evolves to a limiting state as well as various percolation properties of the configuration as a function of time. Disordered systems are a large class of probabilistic models generally motivated by problems arising in condensed matter physics and materials science. First-passage percolation and other closely related models have served as a mathematical model for phenomena as seemingly diverse as properties of randomly porous media (such as aquifers), the growth of cancerous tumors, and the propagation of cracks through brittle material. Ising models capture the basic features of magnetized materials. The proposed work aims at a rigorous understanding of some of the many fundamental mathematical properties of these models.