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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 | import OAI.Probability.InvariantIsing.Pressure.LimitingPressure
import OAI.Probability.InvariantIsing.Spectral.SpectralExtremeLimits
namespace OAI
/-! Theorem 1.1 with the manuscript's exact extreme-eigenvalue hypotheses. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
theorem limiting_pressure_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)
(hcompact : IsCompact (ν : Measure ℝ).support)
(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)) :
Tendsto (fun N => ∫ ω, rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧
∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0))
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) :=
limiting_pressure hhaar hgauss hpub P U hU hHaar eig ν a b hcompact hbound ha hb
(spectral_no_outliers_of_extreme_limits eig a b hmin hmax) hweak
end InvariantIsing
end
end OAI
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