Emergent Equilibrium Structure Along a Critical Cluster Recursion

Shuo Wei; Abbas Ali Saberi; Youjin Deng

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arXiv · 2609.07332

Emergent Equilibrium Structure Along a Critical Cluster Recursion

Shuo Wei·Abbas Ali Saberi·Youjin Deng

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Abstract

Critical universality does not determine the microscopic conditional structure of a probability measure. We study a bicolored cluster recursion constrained to remain critical at every generation, with no equilibrium spin measure or fixed coupling imposed. In both two and three dimensions, the resulting history-dependent sequence develops a common Ising/Fortuin--Kasteleyn (FK) compatibility structure: the second-shell dependence of a one-site conditional law is strongly suppressed, nearest-neighbor effective couplings move progressively toward one another near the critical Ising value, and cluster and interface observables organize around the corresponding FK geometry. An exact cluster-coloring factorization singles out $q=2$ as the point where the residual connectivity weight disappears from the two-color spin marginal. Thus equilibrium-compatible conditional structure can emerge along a trajectory that remains critical throughout.

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Shuo Wei, Abbas Ali Saberi, Youjin Deng. 2026-09-07. Emergent Equilibrium Structure Along a Critical Cluster Recursion. https://arxiv.org/abs/2609.07332

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