Edwards–Sokal coupling and the FK representation

• The Ising model as the two-state Potts model
• The Potts critical temperature and the FK critical parameter
• The q-state Potts model on a finite graph
• The Edwards–Sokal coupling
• Compatible colorings in the Edwards–Sokal coupling
• The Edwards–Sokal coupling with wired boundary conditions
• Ising two-point function via the Edwards–Sokal coupling
• The wired Edwards–Sokal one-point function
• The Ising model in a box as a wired two-state Potts model
• The lattice magnetization as a wired one-point function on the box graph
• Magnetization equals FK percolation probability, through box graphs
• Potts correlations and random-cluster connectivity

DefinitionFinite-volume plus-boundary magnetization at the origin#✓fully proved

The magnetization ⟨σ_0⟩ at the origin under the plus-boundary Ising measure on the box Λ_n at inverse temperature β and zero field.

noncomputable def fvMagnetization (d : ℕ) (β : ℝ) (n : ℕ) : ℝ :=
  ∫ ω, spin ω (origin d) ∂(plusMeasure d n β 0 : Measure (ConfigSpace (Site d)))

Uses 5 · Used by 6

Statement usesConfiguration space of a statistical-mechanics model · Plus-boundary Ising measure in a box · Spin of a configuration at a site · Site of the hypercubic lattice · Origin of the hypercubic lattice

Used byPositive planar percolation at some \(p < 1\) via the Ising model · FK critical point is less than one for \(d \ge 2\), \(q \ge 1\) · Inf-based and sup-based Ising critical points agree for \(d \ge 2\) · Sup-based \(\beta_c\) equals \(-\tfrac{1}{2}\log(1 - p_c)\) for \(d \ge 2\) · Existence of the Ising phase transition · Peierls' theorem

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