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arXiv · 2607.08330

Zero-temperature stochastic Ising model on one-dimensional quasi-transitive graphs

Emilio De Santis
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Abstract

We consider the zero-temperature stochastic Ising process describing $\pm 1$ spin-flip dynamics on an infinite one-dimensional quasi-transitive graph $G=(V,E)$ with finite interaction range $K$. We prove that the zero-temperature limit of the Glauber dynamics for this Ising model exhibits a Type $\mathcal{I}$ behavior (infinite fluctuations of all vertices) if and only if the graph possesses the so-called \emph{shrink property}. For graphs lacking this property, we introduce an algorithmic framework based on an auxiliary spatial automaton to distinguish, in finite time, between Type $\mathcal{F}$ behavior (almost sure local fixation) and Type $\mathcal{M}$ behavior (a mixed regime characterized by the presence of blinkers). We prove that the classification among these three regimes is algorithmically decidable. Furthermore, we provide a constructive example of a graph supporting blinkers of arbitrarily large size.

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Emilio De Santis. 2026-07-09. Zero-temperature stochastic Ising model on one-dimensional quasi-transitive graphs. https://arxiv.org/abs/2607.08330

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