Geodesics in First-Passage Percolation and Random Walks on Critical Percolation Clusters

Abstract

It has been observed that statistical mechanical systems with quenched disorder can exhibit dramatically different properties than their disorder-free counterparts. The introduction of disorder can change the spatial structure of a system's ground state or the typical behavior of the system's free energy (or other quantities of interest). In many systems, the changes to spatial and quantitative behavior are closely linked.

This dissertation explores these themes in two model disordered systems: first-passage percolation and critical percolation. In first-passage percolation, a model for paths in a random potential, the energy of the ground state path of length n is expected to fluctuate like n^chi, and the ground state path is expected to deviate from a straight line by a distance of n^xi. There are longstanding conjectures about the values of chi and xi, and the relation chi = 2 xi - 1 has recently been rigorously established. The prediction that chi < 1/2 (in contrast to a situation where each step of unit length contributed independently to the ground state path's energy) is related to the existence of many ``approximate ground states," and such techniques have proved fruitful for rigorously bounding chi.

In the work presented here, we consider the related issue of wandering of infinite energy-minimizing paths--that is, infinite paths whose energy cannot be reduced by altering any finite segment. One could ask whether such paths tend to have asymptotic direction or remain in sectors. This and related questions have been studied in first-passage percolation and related models; in the setting of first-passage percolation, there are few rigorous results except under strong assumptions. The work presented here develops a framework for studying these questions, and provides the first minimal-assumption results on directional concentration of infinite ground state paths.

The critical percolation systems studied in this dissertation are two related models for the two-dimensional infinite cluster ``at the critical point": the incipient infinite cluster and the invasion percolation cluster. The disorder in these systems gives them a fundamentally different geometry than an ordinary lattice, affecting their transport properties. It was previously shown rigorously that a diffusing particle on the incipient infinite cluster moves strictly slower than a diffusion on the square lattice, on average over the disorder.

The work presented here removes the average over disorder to show a quenched result: for a particular typical realization of the incipient infinite cluster, the diffusion is slow. The result is extended to diffusions on the invasion cluster. The work shows the relationship between the geometrical and transport properties of these models by deriving an upper bound for the speed in terms of critical exponents of the percolation models.