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- W3208616618 abstract "Harmonic activation and transport (HAT) is the Markov chain with the following dynamics. Given the current state, a finite set $U subseteq mathbb{Z}^d$ with at least two elements, HAT transitions to $U cup {y} {setminus}{x}$ by selecting $x$ according to the harmonic measure of $U$, and $y$ according to the probability that random walk, conditioned to hit $U{setminus}{x}$, leaves from $y$ when it first hits $U{setminus}{x}$. In particular, HAT conserves the number of elements in $U$. We are interested in the classification of HAT as recurrent or transient, as the dimension $d$ and number of elements $n$ in the initial state vary. It is known that the stationary distribution of HAT (on certain equivalence classes of states) exists when $d = 2$, for every number of elements $n geq 2$. We prove that HAT exhibits a phase transition in both $d$ and $n$, in the sense that HAT is transient for every $d geq 5$ and $n geq 4$. 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." @default.
- W3208616618 created "2021-11-08" @default.
- W3208616618 creator A5079580858 @default.
- W3208616618 date "2021-10-26" @default.
- W3208616618 modified "2023-09-27" @default.
- W3208616618 title "Existence of a Phase Transition in Harmonic Activation and Transport" @default.
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