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Combinatorial correlation functions in three-dimensional eight-vertex models

Igor G. Korepanov

TL;DR

The paper develops an explicit ${\mathbb F}_2$-linear i-to-${\it t}$ spin transform for a 3D combinatorial eight-vertex model, enabling exact calculation of spin correlations on $2^n\times2^n\times2^n$ blocks under cyclic boundaries. By constructing new thick-space bases and a tensor-product transform, it reduces multi-spin probabilities to tractable Fourier-analytic expressions governed by the eigenstructure of a $3\times3$ matrix $A$ over ${\mathbb F}_2$ (and its transpose). For a concrete $A$, the authors compute single-, pair-, triple-, and quadruple-spin probabilities, revealing mostly independence for small spins but revealing nontrivial 4-spin correlations in structured configurations. The approach generalizes to multiple $A$ choices, preserving key correlation structures, and opens a pathway to rigorous analysis of 3D combinatorial spin systems without integrability assumptions. The results illustrate how algebraic self-similarity and Fourier methods yield exact, scalable insights into complex 3D spin correlation phenomena in a purely combinatorial setting.

Abstract

A new version of the self-similarity spin transform on three-dimensional cubic lattices is proposed that makes possible calculation of nontrivial spin correlations in a "combinatorial" model, in which all permitted spin configurations have equal probabilities.

Combinatorial correlation functions in three-dimensional eight-vertex models

TL;DR

The paper develops an explicit -linear i-to- spin transform for a 3D combinatorial eight-vertex model, enabling exact calculation of spin correlations on blocks under cyclic boundaries. By constructing new thick-space bases and a tensor-product transform, it reduces multi-spin probabilities to tractable Fourier-analytic expressions governed by the eigenstructure of a matrix over (and its transpose). For a concrete , the authors compute single-, pair-, triple-, and quadruple-spin probabilities, revealing mostly independence for small spins but revealing nontrivial 4-spin correlations in structured configurations. The approach generalizes to multiple choices, preserving key correlation structures, and opens a pathway to rigorous analysis of 3D combinatorial spin systems without integrability assumptions. The results illustrate how algebraic self-similarity and Fourier methods yield exact, scalable insights into complex 3D spin correlation phenomena in a purely combinatorial setting.

Abstract

A new version of the self-similarity spin transform on three-dimensional cubic lattices is proposed that makes possible calculation of nontrivial spin correlations in a "combinatorial" model, in which all permitted spin configurations have equal probabilities.
Paper Structure (37 sections, 10 theorems, 72 equations, 4 figures)

This paper contains 37 sections, 10 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction
  2. What this paper is about
  3. The methods used
  4. Remarks on Boltzmann weights
  5. The results
  6. General facts about the linear transformation of spin variables that apply to both its version in block-spin and its new version introduced below in Section \ref{['s:nb']}
  7. Why our new version of spin transform is needed
  8. Correlations calculated in this paper
  9. Contents of the remaining sections
  10. New bases in thick spaces
  11. Terminology: $\bm i$-spins and $\bm t$-spins
  12. Spin transform for a $\bm{2\times 2\times 2}$ block
  13. Spins at the input or output perpendicular to the first coordinate axis
  14. Spins at the inputs or outputs perpendicular to the other coordinate axes
  15. Binary numbering for matrix rows and columns
  16. ...and 22 more sections

Key Result

Proposition 1

$2\times 2\times 2$ block of matrices $A$ (such as in Figure f:b222) acts on rows of vector coordinates in bases new21, nm according to tildes (where, as we remember, $\tilde{A}$ is obtained from $A$ by the Frobenius isomorphism $a_{ij} \mapsto a_{ij}^2$ of all its entries).

Figures (4)

  • Figure 1: $2\times 2\times 2$ block. At each vertex, $3\times 3$ matrix $A$ transforms the row of three input field $F$ elements into the row of three output elements. The positions in a row correspond (in a fixed way) to three edge directions. Field $F=\mathbb F_2$ almost everywhere in this paper
  • Figure 2: LHS: Input of a $2\times 2\times 2$ block, perpendicular to axis 1. Directions of axes 2 and 3 are $\downarrow$ and $\rightarrow$, respectively. RHS: Four separate inputs after switching to basis \ref{['new21']}. Here $\downarrow$ and $\rightarrow$ are directions of the ghost axes $\alpha$ and $\beta$
  • Figure 3: General pictorial representation for a transform of four inputs
  • Figure 4: The second step of transforms in a $4\times 4$ square

Theorems & Definitions (22)

  • Remark
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark
  • Proposition 4
  • proof
  • ...and 12 more