
TL;DR
This paper characterizes the structure of $H$-colorings on high-dimensional discrete tori, revealing long-range correlations and extending entropy-based methods from hypercube cases to more general graphs and weighted models.
Contribution
It provides a structural characterization of $H$-colorings on discrete tori for any graph $H$, generalizing previous work on hypercubes and introducing a weighted coloring model.
Findings
Partition of $H$-colorings into negligible and structured subsets
Long-range correlation phenomenon in high dimensions
Extension of entropy analysis to weighted models
Abstract
For graphs and , an -coloring of is a function from the vertices of to the vertices of that preserves adjacency. -colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of -colorings of the even discrete torus , the graph on vertex set ( even) with two strings adjacent if they differ by 1 (mod ) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes and . In the case the even discrete torus is the discrete hypercube or Hamming cube , the usual nearest neighbor graph on . We obtain, for any and fixed , a structural characterization of the space of -colorings of . We…
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Taxonomy
TopicsFuzzy and Soft Set Theory
