UC Berkeley

Harmonic activation and transport (HAT) is a random process which rearranges a set, one element at a time. More precisely, for integers n ≥ 2 and d ≥ 1, and given an n-element subset U of Zd, HAT is a Markov chain with the following dynamics. HAT removes x from U according to the harmonic measure of x in U, and then adds y according to the probability that a simple random walk from x, conditioned to hit the remaining set, leaves from y when it first does so. This process is then repeated for the resulting set, and so on. We are primarily interested in the classification of HAT as recurrent or transient, as the dimension d and number of elements n in the initial set vary.

Chapter 2 concerns HAT in two dimensions. When d = 2, HAT exhibits a phenomenon we call 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 Z2, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, r? 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 r, up to a constant factor. We prove it is always at least this much, up to an n-dependent factor.

Chapter 3 concerns HAT in higher dimensions. When d ≥ 5 and n ≥ 4, HAT is transient. We prove that, remarkably, transience occurs in only one ``way'': The initial state fragments into clusters of two or three elements—but no other number—which then grow indefinitely separated. We call these clusters dimers and trimers. Underlying this characterization of transience is the fact that, from any state, HAT reaches a state consisting exclusively of dimers and trimers, in a number of steps and with at least a probability which depend on d and n only.

Together, our results establish that HAT exhibits a phase transition in both d and n, in the sense that HAT is positive recurrent when d ≤ 2 or n ≤ 3, but transient when d ≥ 5 and n ≥ 4. Specifically, the phase boundary has a ``corner'': There are d ≥ 3 and n ≥ 4 for which HAT is transient, but HAT is positive recurrent for any smaller d or n.