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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | import OAI.Probability.InvariantIsing.Spectral.ScaledSpectralLaw
namespace OAI
/-! The conditional main theorem at every positive inverse temperature. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
theorem positive_temperature_pressure
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(U : (N : ℕ) → Ω → Orthogonal N) (hU : ∀ N, Measurable (U N))
(hHaar : ∀ N, (P.map (U N)).IsMulRightInvariant)
(eig : (N : ℕ) → Fin N → ℝ) (ν : ProbabilityMeasure ℝ) (a b : ℝ)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε)
(β : ℝ) (hβ : 0 < β)
(hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1)))
atTop (𝓝 ν)) :
Tendsto (fun N => ∫ ω, rotatedPressure (fun i => β*eig N i) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)
atTop (𝓝 (variationalFunctional (measureR (scaledSpectralLaw ν β : Measure ℝ) (β*b))).toReal) ∧
∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (fun i => β*eig N i) (matrixRotation (U N ω)⁻¹) (fun _ => 0))
atTop (𝓝 (variationalFunctional (measureR (scaledSpectralLaw ν β : Measure ℝ) (β*b))).toReal) :=
limiting_pressure hhaar hgauss hpub P U hU hHaar (fun N i => β*eig N i)
(scaledSpectralLaw ν β) (β*a) (β*b)
(scaled_spectral_compact ν a b β hβ.le hbound)
(scaled_spectral_support_bound ν a b β hβ.le hbound)
(scaled_spectral_support_mem ν β a ha) (scaled_spectral_support_mem ν β b hb)
(scaled_spectral_no_outliers eig a b β hβ hno) (scaled_spectral_weak eig ν β hweak)
end InvariantIsing
end
end OAI
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