• Corpus ID: 239885331

Collapse and Diffusion in Harmonic Activation and Transport

  title={Collapse and Diffusion in Harmonic Activation and Transport},
  author={Jacob Calvert and Shirshendu Ganguly and Alan Hammond},
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… 
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Existence of a Phase Transition in Harmonic Activation and Transport
Harmonic activation and transport (HAT) is the Markov chain with the following dynamics. Given the current state, a finite set U ⊆ Z with at least two elements, HAT transitions to U ∪ {y}\{x} by