| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.FreePathExpectation |
| import OAI.Analysis.HardSphere.PathBounds.FreeVisitTravel |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.FreeVisits |
| open HardSphere.Wiener HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital |
| attribute [local irreducible] freeTrajectory pathVisitedSites torusDistance |
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| def singleVisitDefect {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| (η : ℝ) (D : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath 1 L) : ℝ≥0∞ := by |
| classical |
| exact ENNReal.ofReal (Real.exp (η*(if ω∈D then ((pathVisitedSites L (siteCenter L) A 0 ω).card:ℝ) else 0))-1) |
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| lemma singleVisitDefect_measurable {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| (η : ℝ) {D : Set (ConfigurationPath 1 L)} (hD : MeasurableSet D) : |
| Measurable (singleVisitDefect L A η D) := by |
| classical |
| exact ((measurable_const.mul ((pathVisitedSites_measurable L (siteCenter L) A 0).ite hD measurable_const)).exp.sub |
| measurable_const).ennreal_ofReal |
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| lemma singleVisitDefect_reverse {K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) |
| (η : ℝ) {D : Set (ConfigurationPath 1 L)} |
| (hD : ∀ω,reverseConfigurationPath 1 L ω∈D ↔ ω∈D) (ω : ConfigurationPath 1 L) : |
| singleVisitDefect L A η D (reverseConfigurationPath 1 L ω)=singleVisitDefect L A η D ω := by |
| classical |
| simp only [singleVisitDefect,hD,pathVisitedSites_reverse] |
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| lemma free_single_defect_bound {η₀ : ℝ} (hη₀ : 0≤η₀) : ∃ C : ℝ, 0<C ∧ |
| ∀ {K : ℕ} [NeZero K] (L : Length), L.val=(K:ℝ) → ∀ A : Finset (DiscreteTorus K), |
| ∀ {t η ε : ℝ}, 0<t → t≤2 → 0≤η → η≤η₀ → 0≤ε → |
| ∀ D : Set (ConfigurationPath 1 L), MeasurableSet D → |
| (∀ y : Torus L, coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈D}≤ENNReal.ofReal ε) → |
| ∀ y : Torus L, |
| oneFreeExpectation t (singleVisitDefect L A η D) y ≤ |
| ENNReal.ofReal (C*η*ε^(1/3:ℝ)*Real.exp (-spatialSiteDistance L (A.image (siteCenter L)) y)) := by |
| classical |
| obtain ⟨C,hC,hbound⟩ := free_visit_bound hη₀ |
| refine ⟨C.toReal+1,by positivity,?_⟩ |
| intro K _ L hL A t η ε ht ht2 hη hηη hε D hD hprob y |
| let E := {w : CoordinatePaths 1 | freeTrajectory 1 L t w (oneConfiguration y)∈D} |
| have hE : MeasurableSet E := hD.preimage |
| ((freeTrajectory_continuous 1 L t).measurable.comp (measurable_id.prodMk measurable_const)) |
| have H := hbound L hL A 0 ht ht2 hη hηη (oneConfiguration y) E hE (ENNReal.ofReal ε) (hprob y) |
| have heq : oneFreeExpectation t (singleVisitDefect L A η D) y = |
| ∫⁻ w,visitDefect L A 0 t η (oneConfiguration y) E w ∂coordinateWiener 1 := by |
| rfl |
| rw [heq] |
| apply H.trans |
| have hC' : C≤ENNReal.ofReal (C.toReal+1) := by |
| calc |
| C = ENNReal.ofReal C.toReal := (ENNReal.ofReal_toReal hC.ne).symm |
| _ ≤ ENNReal.ofReal (C.toReal+1) := ENNReal.ofReal_le_ofReal (by linarith) |
| calc |
| C*ENNReal.ofReal η*ENNReal.ofReal ε^(1/3:ℝ)* |
| ENNReal.ofReal (Real.exp (-visitStartDistance L A 0 (oneConfiguration y))) ≤ |
| ENNReal.ofReal (C.toReal+1)*ENNReal.ofReal η*ENNReal.ofReal ε^(1/3:ℝ)* |
| ENNReal.ofReal (Real.exp (-visitStartDistance L A 0 (oneConfiguration y))) := by gcongr |
| _ = _ := by |
| rw [ENNReal.ofReal_rpow_of_nonneg hε (show 0≤(1/3:ℝ) by norm_num), |
| ←ENNReal.ofReal_mul (show 0≤C.toReal+1 by positivity), |
| ←ENNReal.ofReal_mul (show 0≤(C.toReal+1)*η by positivity), |
| ←ENNReal.ofReal_mul (show 0≤(C.toReal+1)*η*ε^(1/3:ℝ) by positivity)] |
| rfl |
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|
| end HardSphere.FreeVisits |
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| end |
| end |
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| end |
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| end OAI |
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