| import OAI.Probability.InvariantIsing.Pressure.GroundStateLimit |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExtremeLimits |
|
|
| namespace OAI |
|
|
| /-! The ground-state corollary under the manuscript's exact extreme-eigenvalue hypotheses. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
|
|
| namespace InvariantIsing |
|
|
| theorem ground_state_limit_of_extreme_limits |
| (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) |
| (hmin : Tendsto (fun k => spectralMinimum (eig (k+1))) atTop (𝓝 a)) |
| (hmax : Tendsto (fun k => spectralMaximum (eig (k+1))) atTop (𝓝 b)) |
| (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) |
| atTop (𝓝 ν)) : |
| ∃ M : ℝ, |
| Tendsto (fun N => ∫ ω, groundStateEnergy (eig N) (matrixRotation (U N ω)⁻¹) ∂P) |
| atTop (𝓝 M) ∧ |
| (∀ᵐ ω ∂P, Tendsto (fun N => groundStateEnergy (eig N) (matrixRotation (U N ω)⁻¹)) |
| atTop (𝓝 M)) ∧ |
| Tendsto (thermalVariationalValue ν b) atTop (𝓝 M) := |
| ground_state_limit hhaar hgauss hpub P U hU hHaar eig ν a b hbound ha hb |
| (spectral_no_outliers_of_extreme_limits eig a b hmin hmax) hweak |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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|
