lean/OAI/Probability/InvariantIsing/Unconditional.lean

import OAI.Probability.InvariantIsing.Haar.HaarConcentrationInput
import OAI.Probability.InvariantIsing.Gaussian.GaussianVarianceInput
import OAI.Probability.InvariantIsing.Fields.FieldPairInput
import OAI.Probability.InvariantIsing.Magnetic.RestrictedFieldPairInput
import OAI.Probability.InvariantIsing.Gaussian.MarchenkoPasturInput
import OAI.Probability.InvariantIsing.Gaussian.DavidsonSzarekInput
import OAI.Probability.InvariantIsing.Pressure.MainPressure
import OAI.Probability.InvariantIsing.Magnetic.MagneticWassersteinMain
import OAI.Probability.InvariantIsing.Pressure.PositiveTemperaturePressure
import OAI.Probability.InvariantIsing.Pressure.GroundStateLimit
import OAI.Probability.InvariantIsing.Pressure.RandomPressureMain
import OAI.Probability.InvariantIsing.Pressure.RandomPressureAlmostSure
import OAI.Probability.InvariantIsing.Pressure.RandomFieldAlmostSure
import OAI.Probability.InvariantIsing.Gaussian.GaussianPatternPressureLimit
import OAI.Probability.InvariantIsing.Gaussian.GaussianPatternGroundLimit
import OAI.Probability.InvariantIsing.Gaussian.GaussianPatternCorollary

namespace OAI

/-! Limiting invariant Ising pressure, magnetic fields, and Gaussian patterns. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
universe u
namespace InvariantIsing

theorem limiting_pressure_of_extreme_limits_unconditional :
  ∀ {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_of_extreme_limits haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved

theorem limiting_wasserstein_field_pressure_of_extreme_limits_unconditional :
  ∀ {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))
    (field : (N : ℕ) → Fin N → ℝ) (ξ : ProbabilityMeasure ℝ)
    (_hξ : Integrable (fun x : ℝ => x) (ξ : Measure ℝ))
    (_hw : Tendsto (fun k => fieldWassersteinOne (empiricalSpectralLaw (Nat.succ_pos k) (field (k+1))) ξ)
      atTop (𝓝 0)),
    Tendsto (fun N => ∫ ω, rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (field N) ∂P)
      atTop (𝓝 (magneticFieldFunctional (ν : Measure ℝ) b ξ)) ∧
    ∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (field N))
      atTop (𝓝 (magneticFieldFunctional (ν : Measure ℝ) b ξ)) :=
  @limiting_wasserstein_field_pressure_of_extreme_limits haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandRestrictedFieldPair_proved

theorem positive_temperature_pressure_unconditional :
  ∀ {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) :=
  @positive_temperature_pressure haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved

theorem ground_state_limit_unconditional :
  ∀ {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)
    (_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 haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved

theorem random_pressure_tendsto_in_measure_unconditional
    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (eig : (N : ℕ) → Ω → Fin N → ℝ) (Y : ℕ → Ω → ℝ)
    (heig : ∀ N, Measurable (eig N)) (hY : ∀ N, Measurable (Y N))
    (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
    [∀ N, (H N).IsMulRightInvariant]
    (hlaw : ∀ N, ConditionalFieldOrbitLaw P (fun ω => (eig N ω,fun _ => 0)) (Y N) (H N))
    (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)
    (hweak : TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure
      (empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω))) atTop
      (fun _ => LevyProkhorov.ofMeasure ν))
    (hexcess : TendstoInMeasure P (fun k ω => spectralExcess (eig (k+1) ω) a b)
      atTop (fun _ => 0)) :
    TendstoInMeasure P (fun k => Y (k+1)) atTop
      (fun _ => (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) :=
  random_pressure_tendsto_in_measure haarConcentration_proved gaussianLipschitzVariance_proved
    panchenkoTalagrandFieldPair_proved P eig Y heig hY H hlaw ν a b hcompact hbound ha hb
    hweak hexcess

theorem random_pressure_limit_unconditional :
  ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (eig : (N : ℕ) → Ω → Fin N → ℝ) (Y : ℕ → Ω → ℝ)
    (_heig : ∀ N, Measurable (eig N)) (_hY : ∀ N, Measurable (Y N))
    (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
    [∀ N, (H N).IsMulRightInvariant]
    (_hlaw : ∀ N, ConditionalFieldOrbitLaw P (fun ω => (eig N ω,fun _ => 0)) (Y N) (H N))
    (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)
    (_hweak : TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure
      (empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω))) atTop
      (fun _ => LevyProkhorov.ofMeasure ν))
    (_hexcess : TendstoInMeasure P (fun k ω => spectralExcess (eig (k+1) ω) a b)
      atTop (fun _ => 0))
    (_hUI : UniformIntegrable (fun k => Y (k+1)) 1 P),
    TendstoInMeasure P (fun k => Y (k+1)) atTop
      (fun _ => (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧
    Tendsto (fun k => eLpNorm (fun ω => Y (k+1) ω-
      (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) 1 P) atTop (𝓝 0) ∧
    Tendsto (fun k => ∫ ω, Y (k+1) ω ∂P) atTop
      (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) :=
  @random_pressure_limit haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved

theorem random_pressure_tendsto_ae_unconditional :
  ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (eig : (N : ℕ) → Ω → Fin N → ℝ) (Y : ℕ → Ω → ℝ)
    (_heig : ∀ N, Measurable (eig N)) (_hY : ∀ N, Measurable (Y N))
    (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
    [∀ N, (H N).IsMulRightInvariant]
    (_hlaw : ∀ N, ConditionalFieldOrbitLaw P (fun ω => (eig N ω,fun _ => 0)) (Y N) (H N))
    (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)
    (_hno : ∀ᵐ ω ∂P, ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop,
      ∀ i, a-ε ≤ eig N ω i ∧ eig N ω i ≤ b+ε)
    (_hweak : ∀ᵐ ω ∂P, Tendsto (fun k =>
      empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω)) atTop (𝓝 ν)),
    ∀ᵐ ω ∂P, Tendsto (fun N => Y N ω) atTop
      (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) :=
  @random_pressure_tendsto_ae haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved

theorem random_field_pressure_tendsto_ae_unconditional :
  ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (d : (N : ℕ) → Ω → FieldSpectralData N) (Y : ℕ → Ω → ℝ)
    (_hd : ∀ N, Measurable (d N)) (_hY : ∀ N, Measurable (Y N))
    (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
    [∀ N, (H N).IsMulRightInvariant]
    (_hlaw : ∀ N, ConditionalFieldOrbitLaw P (d N) (Y N) (H N))
    (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)
    (_hno : ∀ᵐ ω ∂P, ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop,
      ∀ i, a-ε ≤ (d N ω).1 i ∧ (d N ω).1 i ≤ b+ε)
    (_hweak : ∀ᵐ ω ∂P, Tendsto (fun k =>
      empiricalSpectralLaw (Nat.succ_pos k) (d (k+1) ω).1) atTop (𝓝 ν))
    (ξ : ProbabilityMeasure ℝ) (_hξ : Integrable (fun x : ℝ => x) (ξ : Measure ℝ))
    (_hw : ∀ᵐ ω ∂P, Tendsto (fun k => fieldWassersteinOne
      (empiricalSpectralLaw (Nat.succ_pos k) (d (k+1) ω).2) ξ) atTop (𝓝 0)),
    ∀ᵐ ω ∂P, Tendsto (fun N => Y N ω) atTop
      (𝓝 (magneticFieldFunctional (ν : Measure ℝ) b ξ)) :=
  @random_field_pressure_tendsto_ae haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandRestrictedFieldPair_proved

theorem gaussianPattern_pressure_limit_unconditional :
  ∀ (α : ℝ) (_hα : 0 < α)
    {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N)))
    (_hZ : ∀ N, Measurable (Z N)) (_hlaw : ∀ N, HasLaw (Z N) (stdGaussian _) P)
    (c : ℝ),
    TendstoInMeasure P (fun k ω => gaussianPatternPressure (N := k+1) (m := gaussianPatternCount α (k+1)) c (Z (k+1) ω)) atTop
      (fun _ => gaussianPatternLimit α c) ∧
    Tendsto (fun k => eLpNorm (fun ω => gaussianPatternPressure (N := k+1) (m := gaussianPatternCount α (k+1)) c (Z (k+1) ω)-
      gaussianPatternLimit α c) 1 P) atTop (𝓝 0) ∧
    Tendsto (fun k => ∫ ω, gaussianPatternPressure (N := k+1) (m := gaussianPatternCount α (k+1)) c (Z (k+1) ω) ∂P) atTop
      (𝓝 (gaussianPatternLimit α c)) :=
  fun α hα => @gaussianPattern_pressure_limit haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved α hα (marchenkoPastur_proved α hα) (davidsonSzarek_proved α hα)

theorem gaussianPattern_ground_limit_unconditional :
  ∀ (α : ℝ) (_hα : 0 < α)
    {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N)))
    (_hZ : ∀ N, Measurable (Z N)) (_hlaw : ∀ N, HasLaw (Z N) (stdGaussian _) P)
    (ε : ℝ),
    ∃ e : ℝ,
      TendstoInMeasure P (fun k ω => gaussianPatternGroundEnergy ε (Z (k+1) ω)) atTop (fun _ => e) ∧
      Tendsto (fun k => eLpNorm (fun ω => gaussianPatternGroundEnergy ε (Z (k+1) ω)-e) 1 P)
        atTop (𝓝 0) ∧
      Tendsto (fun k => ∫ ω, gaussianPatternGroundEnergy ε (Z (k+1) ω) ∂P) atTop (𝓝 e) ∧
      Tendsto (fun β => gaussianPatternLimit α (ε*β)/β) atTop (𝓝 e) :=
  fun α hα => @gaussianPattern_ground_limit haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved α hα (marchenkoPastur_proved α hα) (davidsonSzarek_proved α hα)

theorem gaussianPattern_norm_limits_unconditional :
  ∀ (α : ℝ) (_hα : 0 < α)
    {Ω : Type u} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
    (Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N)))
    (_hZ : ∀ N, Measurable (Z N)) (_hlaw : ∀ N, HasLaw (Z N) (stdGaussian _) P),
    ∃ eplus eminus : ℝ, 0 ≤ eplus ∧ eminus ≤ 0 ∧
      Tendsto (fun β => gaussianPatternLimit α β/β) atTop (𝓝 eplus) ∧
      Tendsto (fun β => gaussianPatternLimit α (-β)/β) atTop (𝓝 eminus) ∧
      (TendstoInMeasure P (fun k ω => ‖gaussianPatternOperator (Z (k+1) ω)‖/(k+1)) atTop
        (fun _ => Real.sqrt (2*eplus)) ∧
       Tendsto (fun k => eLpNorm (fun ω => ‖gaussianPatternOperator (Z (k+1) ω)‖/(k+1)-
        Real.sqrt (2*eplus)) 1 P) atTop (𝓝 0)) ∧
      (TendstoInMeasure P (fun k ω =>
        (Finset.univ.inf' Finset.univ_nonempty (fun σ : Spin (k+1) => ‖gaussianPatternSum (Z (k+1) ω) σ‖))/(k+1))
        atTop (fun _ => Real.sqrt (-2*eminus)) ∧
       Tendsto (fun k => eLpNorm (fun ω =>
        (Finset.univ.inf' Finset.univ_nonempty (fun σ : Spin (k+1) => ‖gaussianPatternSum (Z (k+1) ω) σ‖))/(k+1)-
          Real.sqrt (-2*eminus)) 1 P) atTop (𝓝 0)) :=
  fun α hα => @gaussianPattern_norm_limits haarConcentration_proved gaussianLipschitzVariance_proved panchenkoTalagrandFieldPair_proved α hα (marchenkoPastur_proved α hα) (davidsonSzarek_proved α hα)

end InvariantIsing

end

end OAI