Emergent Equilibrium Structure Along a Critical Cluster Recursion
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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BibTeX
@misc{wei2026emergent,
title = {{Emergent Equilibrium Structure Along a Critical Cluster Recursion}},
author = {Shuo Wei and Abbas Ali Saberi and Youjin Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07332},
eprint = {2609.07332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07332},
}
RIS
TY - UNPB
TI - Emergent Equilibrium Structure Along a Critical Cluster Recursion
AU - Shuo Wei
AU - Abbas Ali Saberi
AU - Youjin Deng
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07332
UR - https://arxiv.org/abs/2609.07332
AB - 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.
ER -