Collapse and Diffusion in Harmonic Activation and Transport

TL;DR

This paper introduces the Harmonic Activation and Transport (HAT) process on subsets of the integer lattice, revealing a collapse phenomenon where the set's diameter shrinks logarithmically, and establishes its stationary distribution and convergence to Brownian motion.

Contribution

The paper defines HAT, proves the collapse phenomenon, characterizes the stationary distribution, and analyzes extremal harmonic measure and escape probabilities in two-dimensional lattice sets.

Findings

Diameter shrinks to logarithm over steps proportional to log n

Stationary distribution exists with exponential tightness of diameter

Center of mass converges to Brownian motion

Abstract

For an $n$-element subset $U$ of $\mathbb{Z}^2$, select $x$ from $U$ according to harmonic measure from infinity, remove $x$ from $U$, and start a random walk from $x$. If the walk leaves from $y$ when it first enters $U$, add $y$ to $U$. Iterating this procedure constitutes the process we call Harmonic Activation and Transport (HAT). HAT exhibits a phenomenon we refer to as collapse: informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion. To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among $n$-element subsets of $\mathbb{Z}^2$, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, $d$? Concerning the former, examples abound for which the harmonic measure is exponentially small in $n$. We prove that it can be no smaller than exponential in $n \log n$. Regarding the latter, the escape probability is at most the reciprocal of $\log d$, up to a constant factor. We prove it is always at least this much, up to an $n$-dependent factor.