| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.LogPartition |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
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| theorem rotation_logPartition_tail {n : ℕ} (hn : 0 < n) (β θ u : ℝ) (hu : 0 ≤ u) : |
| (disorderPairLaw n).real {p | u ≤ logPartition β (rotateDisorder θ p).1- |
| logPartition β p.1} ≤ Real.exp (-u^2/(θ^2*β^2*(n:ℝ))) := by |
| let v : NNReal := ⟨θ^2*β^2*(n:ℝ)/2,by positivity⟩ |
| have HS : HasSubgaussianMGF (fun p : Disorder n × Disorder n => |
| logPartition β (rotateDisorder θ p).1-logPartition β p.1) v (disorderPairLaw n) := by |
| refine ⟨rotation_exp_integrable hn β θ,fun t => ?_⟩ |
| apply (rotation_exp_integral_le hn β θ t).trans |
| apply Real.exp_le_exp.mpr |
| change t^2*θ^2*β^2*((n:ℝ)-1)/4 ≤ (θ^2*β^2*(n:ℝ)/2)*t^2/2 |
| have H : 0 ≤ t^2*θ^2*β^2 := by positivity |
| nlinarith |
| convert HS.measure_ge_le hu using 1 |
| congr 2 |
| change θ^2*β^2*(n:ℝ) = 2*(θ^2*β^2*(n:ℝ)/2) |
| ring |
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| theorem product_event_lower_of_sections {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {A : Set X} {E : Set (X × Y)} (hA : MeasurableSet A) (hE : MeasurableSet E) |
| {c : ℝ} (hc : 0 ≤ c) (hsec : ∀ x ∈ A, c ≤ ν.real {y | (x,y) ∈ E}) : |
| c*μ.real A ≤ (μ.prod ν).real E := by |
| have H : ENNReal.ofReal c*μ A ≤ (μ.prod ν) E := by |
| rw [Measure.prod_apply hE,← lintegral_indicator_const hA (ENNReal.ofReal c)] |
| apply lintegral_mono |
| intro x |
| by_cases hx : x ∈ A |
| · rw [Set.indicator_of_mem hx] |
| exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top ν _)).mpr (hsec x hx) |
| · rw [Set.indicator_of_notMem hx] |
| exact bot_le |
| have HT := ENNReal.toReal_mono (measure_ne_top (μ.prod ν) E) H |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal hc,measureReal_def] using HT |
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| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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