| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.PressureGapProbability |
| import OAI.Probability.SKBarriers.Coverage.FiniteRateAsymptotic |
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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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|
| def restrictedPressure {n d : ℕ} (β : ℝ) (S : Finset (ReplicaConfig n d)) : ℝ := |
| (∫ J, restrictedLogPartition β S J ∂disorderLaw n)/(n:ℝ) |
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| theorem replicaGibbsMass_integral_le_of_finiteInf_gap {n d : ℕ} (hn : 0 < n) (hd : 0 < d) |
| {β η e : ℝ} (hβ : β ≠ 0) (hη : 0 ≤ η) |
| {S : Finset (ReplicaConfig n d)} (hS : S.Nonempty) |
| (hlower : finiteParisiInf β-e ≤ quenchedPressure β n) |
| (herror : 2*(d:ℝ)*e ≤ η) |
| (hpressure : restrictedPressure β S ≤ (d:ℝ)*finiteParisiInf β-η) : |
| (∫ J, replicaGibbsMass β J S ∂disorderLaw n) ≤ |
| Real.exp (-(n:ℝ)*η/4)+2*Real.exp (-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2)) := by |
| apply replicaGibbsMass_integral_le_of_mean_gap hn hd hβ hη hS |
| have hn0 : 0 < (n:ℝ) := by exact_mod_cast hn |
| have hd0 : 0 ≤ (d:ℝ) := by positivity |
| have H := mul_le_mul_of_nonneg_left hlower hd0 |
| have HP : restrictedPressure β S-(d:ℝ)*quenchedPressure β n ≤ -η/2 := by linarith |
| unfold restrictedPressure quenchedPressure at HP |
| have HN := mul_le_mul_of_nonneg_right HP hn0.le |
| have he : ((∫ J, restrictedLogPartition β S J ∂disorderLaw n)/(n:ℝ)- |
| (d:ℝ)*((∫ J, logPartition β J ∂disorderLaw n)/(n:ℝ)))*(n:ℝ) = |
| (∫ J, restrictedLogPartition β S J ∂disorderLaw n)- |
| (d:ℝ)*(∫ J, logPartition β J ∂disorderLaw n) := by field_simp [ne_of_gt hn0] |
| rw [he] at HN |
| linarith |
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| theorem eventual_replicaGibbsMass_pressure_gap {β : ℝ} (hβ : 0 < β) (d : ℕ) (hd : 0 < d) : |
| ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ (S : Finset (ReplicaConfig n d)) (η : ℝ), |
| S.Nonempty → 0 ≤ η → 2*(d:ℝ)*C*(n:ℝ)^(-(1:ℝ)/24) ≤ η → |
| restrictedPressure β S ≤ (d:ℝ)*finiteParisiInf β-η → |
| (∫ J, replicaGibbsMass β J S ∂disorderLaw n) ≤ |
| Real.exp (-(n:ℝ)*η/4)+2*Real.exp (-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2)) := by |
| obtain ⟨C,hC,HC⟩ := quenchedPressure_quantitative_finiteInf hβ |
| refine ⟨C,hC,?_⟩ |
| filter_upwards [HC,eventually_gt_atTop (0:ℕ)] with n hn hn0 |
| intro S η hS hη herr hpress |
| apply replicaGibbsMass_integral_le_of_finiteInf_gap hn0 hd hβ.ne' hη hS hn.1 _ hpress |
| simpa only [mul_assoc] using herr |
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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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