| import Mathlib |
| import OAI.Probability.Perceptron.Interpolation.ReplicaMeanConst |
| import OAI.Probability.Perceptron.Cascade.ReweightedMomentComparison |
| import OAI.Probability.Perceptron.Cavity.CavityBoundedContinuous |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma expectedPressure_terminal_scaling (α β : ℝ) (φ : ℝ→ᵇℝ) (n : ℕ) : |
| expectedPressure α β φ (n+1)=expectedPressure α 1 (β • φ) (n+1) := by |
| unfold expectedPressure |
| apply integral_congr_ae |
| filter_upwards [] with g |
| rw [←normalizedPatternLog_source α β φ n g, |
| ←normalizedPatternLog_source α 1 (β • φ) n g,one_smul] |
|
|
| theorem main (P : Measure BrownianPath) [IsProbabilityMeasure P] |
| (hB : IsBrownianReal brownianEval P) |
| (α β : ℝ) (hα : 0 < α) (_hβ : 0 < β) (φ : ℝ →ᵇ ℝ) : |
| ∃ p : ℝ, variationalValue P α β φ = (p : EReal) ∧ |
| Tendsto (fun N : ℕ => expectedPressure α β φ (N + 1)) atTop (𝓝 p) ∧ |
| ∀ ε : ℝ, 0 < ε → |
| Tendsto (fun N : ℕ => patternLaw α (N + 1) |
| {g | ε < |pressure α β φ (N + 1) g - p|}) atTop (𝓝 0) := by |
| let p:=(terminalVariational P α (β • φ)).toReal |
| have hp : Tendsto (fun N : ℕ=>expectedPressure α β φ (N+1)) atTop (𝓝 p) := by |
| have heq : (fun N : ℕ=>expectedPressure α β φ (N+1)) = |
| (fun N : ℕ=>expectedPressure α 1 (β • φ) (N+1)) := |
| funext (expectedPressure_terminal_scaling α β φ) |
| rw [heq] |
| exact expectedPressure_limit_terminal P hB hα (β • φ) |
| refine ⟨p,?_,hp,?_⟩ |
| · rw [variationalValue_eq_terminal] |
| exact (terminalVariational_coe_toReal P α hα.le (β • φ)).symm |
| · exact pressure_probability_of_expected_limit α β hα.le φ p hp |
|
|
| end SphericalPerceptronFreeEnergy |
|
|
| end |
|
|
| end OAI |
|
|
