High-temperature expansion of the Ising partition function

DefinitionZero-field Ising partition function with unit couplings#✓fully proved

The zero-field Ising partition function of G with unit couplings at inverse temperature β.

noncomputable def ons_Z (β : ℝ) : ℝ := isingZ G β 0

Uses 1 · Used by 7

Statement usesIsing partition function

Used byHigh-temperature expansion · Finite-volume pressure · Finite-volume pressure decomposition for \(\beta \ge 0\) · Onsager's formula from the spin Kac-Ward identities · High-temperature expansion · Pressure decomposition on the \(L \times L\) torus for \(\beta \ge 0\) · Onsager's formula

DefinitionEven-subgraph generating function of a graph#✓fully proved

The even-subgraph generating function ∑_F x^|F| of G, over edge sets F in which every vertex has even degree.

noncomputable def ons_X (x : ℝ) : ℝ :=
  ∑ F ∈ G.edgeFinset.powerset.filter IsEvenSubgraph, x ^ F.card

Uses 2 · Used by 7

Statement usesEven subgraph condition on an edge set · Number of edges of an edge set incident to a vertex

Used byHigh-temperature expansion · Limit statement for \(\log X_L(\tanh\beta)/L^2\) on the torus · Finite-volume pressure decomposition for \(\beta \ge 0\) · Onsager's formula from the spin Kac-Ward identities · High-temperature expansion · Pressure decomposition on the \(L \times L\) torus for \(\beta \ge 0\) · Onsager's formula

TheoremHigh-temperature expansion#✓fully proved

High-temperature expansion (van der Waerden 1941). Z(β) = 2^|V| (cosh β)^|E| X(tanh β) for the zero-field Ising model on a finite graph (B. L. van der Waerden, Die lange Reichweite der regelmässigen Atomanordnung in Mischkristallen, 1941).

theorem ons_hte (β : ℝ) :
    ons_Z G β
      = (2 : ℝ) ^ Fintype.card V * (Real.cosh β) ^ G.edgeFinset.card
        * ons_X G (Real.tanh β)

Uses 4 · Used by 5

Statement usesEven-subgraph generating function of a graph · Zero-field Ising partition function with unit couplings

Proof usesIsing partition function · High-temperature expansion

Used byFinite-volume pressure decomposition for \(\beta \ge 0\) · Onsager's formula from the spin Kac-Ward identities · High-temperature expansion · Pressure decomposition on the \(L \times L\) torus for \(\beta \ge 0\) · Onsager's formula

3 fully proved

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