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dd14c26 | 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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 | 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
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