The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.

arXiv:1012.1485Europhysics Letters (EPL)Dec 1, 201026 citations

Percolation Theory Determines Topological Color Code Thresholds on Hexagonal and Square-Octagonal Lattices — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): $p_c = 1 - 2\sin(\pi/18) \approx 0.6527$ - For square-octagonal lattice: $p_c \approx 0.6764$ (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): $\approx 0.6527 / 0.6764 \approx 0.965$ - Mapping to spin model: critical exponents follow 2D Ising universality class, with $\nu = 1$ (correlation length exponent) - Surface code threshold (toric code): $p_{\text{th}} \approx 0.103$ (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group $D_6$ ), bond angles $120^\circ$ — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): $ p_c = 1 - 2\sin(\pi/18) \approx 0.6527 $ - For square-octagonal lattice: $ p_c \approx 0.6764 $ (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): $ \approx 0.6527 / 0.6764 \approx 0.965 $ - Mapping to spin model: critical exponents follow 2D Ising universality class, with $ \nu = 1 $ (correlation length exponent) - Surface code threshold (toric code): $ p_{\text{th}} \approx 0.103 $ (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group $ D_6 $), bond angles $ 120^\circ $ — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 28, 20260 citations

Cluster percolation and pseudocritical behaviour in spin models

Santo Fortunato, Helmut Satz

arXiv:hep-lat/0103010Physics Letters BJun 1, 200119 citations

Cluster percolation and explicit symmetry breaking in spin models

Santo Fortunato, Helmut Satz

arXiv:hep-lat/0109018Nuclear Physics B - Proceedings SupplementsMar 1, 20026 citations

p_c = 1 - 2\sin(\pi/18) \approx 0.6527

- For square-octagonal lattice:

p_c \approx 0.6764

(exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal):

\approx 0.6527 / 0.6764 \approx 0.965

- Mapping to spin model: critical exponents follow 2D Ising universality class, with

\nu = 1

(correlation length exponent) - Surface code threshold (toric code):

p_{\text{th}} \approx 0.103

(for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group

D_6

), bond angles

120^\circ

— related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com