| import Mathlib |
| import OAI.Analysis.HardSphere.PathBounds.InteractingVisits |
|
|
| namespace OAI |
|
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| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.ShiftVisits |
| open HardSphere.Wiener HardSphere.Lattice HardSphere.CellOrbital HardSphere.BathSlice HardSphere.FreeVisits |
| attribute [local irreducible] freeTrajectory |
| def defectiveTotalVisits {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| (D : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath n L) : ℕ := by |
| classical |
| exact ∑ i, if onePathAt i ω∈D then (pathVisitedSites L (center L c) A i ω).card else 0 |
|
|
| lemma pathVisitedSites_onePathAt {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| (i : Fin n) (ω : ConfigurationPath n L) : |
| pathVisitedSites L (center L c) A 0 (onePathAt i ω)=pathVisitedSites L (center L c) A i ω := by |
| simp only [pathVisitedSites,pathVisitsBall,onePathAt,oneConfiguration,ContinuousMap.coe_mk] |
| rfl |
|
|
| lemma exp_defectiveTotalVisits {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| (D : Set (ConfigurationPath 1 L)) {η : ℝ} (hη : 0≤η) (ω : ConfigurationPath n L) : |
| ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A D ω : ℝ)))= |
| ∏ i, (1+singleDefect L c A η D (onePathAt i ω)) := by |
| classical |
| simp only [defectiveTotalVisits,Nat.cast_sum,Finset.mul_sum,Real.exp_sum, |
| ENNReal.ofReal_prod_of_nonneg (fun _ _ ↦ (Real.exp_pos _).le)] |
| apply Finset.prod_congr rfl |
| intro i _ |
| rw [singleDefect,pathVisitedSites_onePathAt] |
| have hx : 0≤η*(if onePathAt i ω∈D then ((pathVisitedSites L (center L c) A i ω).card:ℝ) else 0) := by positivity |
| have he : 0≤Real.exp (η*(if onePathAt i ω∈D then ((pathVisitedSites L (center L c) A i ω).card:ℝ) else 0))-1 := sub_nonneg.mpr (Real.one_le_exp_iff.mpr hx) |
| simp only [Nat.cast_ite,Nat.cast_zero] |
| rw [←ENNReal.ofReal_one,←ENNReal.ofReal_add zero_le_one he] |
| congr 1 |
| ring |
|
|
| theorem interacting_visit_mgf {η₀ : ℝ} (hη₀ : 0≤η₀) : |
| ∃ Cdel Cfree Csp αstar : ℝ, |
| 0<Cdel ∧ 0<Cfree ∧ 0<Csp ∧ 0<αstar ∧ |
| ∀ (n K : ℕ) [NeZero K] (L : Length), L.val=(K:ℝ) → ∀ r D : ℝ, |
| D=(n:ℝ)/L.val^3 → 1≤D → αstar≤D*r → 0<r → |
| (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ (g : Jet n L), IsGroundVector n L r g → |
| ∀ (c : Torus L) (A : Finset (DiscreteTorus K)) {t η ε : ℝ}, |
| 0<t → t≤2 → 0≤η → η≤η₀ → 0≤ε → |
| ∀ E : Set (ConfigurationPath 1 L), MeasurableSet E → |
| (∀ω,reverseConfigurationPath 1 L ω∈E ↔ ω∈E) → |
| (∀ y : Torus L, coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈E}≤ENNReal.ofReal ε) → |
| Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ))≤1 → |
| (∫⁻ ω, ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ))) |
| ∂groundPathLaw n L r t g) ≤ |
| ENNReal.ofReal (Real.exp (Csp*(Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ)))*D*(A.card:ℝ))) := by |
| classical |
| obtain ⟨Cdel,hCdel,hdel⟩ := ground_product_one_add |
| obtain ⟨Cfree,hCfree,hfree⟩ := free_single_defect_bound hη₀ |
| obtain ⟨Csp,αstar,hCsp,hα,hsp⟩ := spatial_count_mgf_enn |
| refine ⟨Cdel,Cfree,Csp,αstar,hCdel,hCfree,hCsp,hα,?_⟩ |
| intro n K inst L hL r D hDeq hD hαr hr hs g hg c A t η ε ht ht2 hη hη0 hε E hE hrev hprob hz1 |
| let z := Real.exp (t*Cdel*D*r)*(Cfree*η*ε^(1/3:ℝ)) |
| have hz : 0≤z := by positivity |
| have hsize : 1≤L.val := by rw [hL]; exact_mod_cast (NeZero.pos K) |
| let := hg.groundPathLaw_probability ht.le |
| by_cases hA : A.Nonempty |
| · let B := A.image (center L c) |
| have hB : B.Nonempty := hA.image _ |
| have hDexp : t*Cdel*(n:ℝ)*r/L.val^3=t*Cdel*D*r := by rw [hDeq]; ring |
| calc |
| _ = ∫⁻ ω,∏i,(1+singleDefect L c A η E (onePathAt i ω)) ∂groundPathLaw n L r t g := |
| lintegral_congr (exp_defectiveTotalVisits L c A E hη) |
| _ ≤ ∫⁻ X,∏i,(1+ENNReal.ofReal (Real.exp (t*Cdel*(n:ℝ)*r/L.val^3))* |
| oneFreeExpectation t (singleDefect L c A η E) (X i)) ∂groundPositionLaw n L g := |
| hdel n L r t hr ht hs g hg _ (singleDefect_measurable L c A η hE) (singleDefect_reverse L c A η hrev) |
| _ ≤ ∫⁻ X,ENNReal.ofReal (Real.exp (z*weightedSiteCount L B X)) ∂groundPositionLaw n L g := by |
| apply lintegral_mono |
| intro X |
| apply le_trans _ (product_one_add_exp_le L B hB hz X) |
| apply Finset.prod_le_prod |
| intro i _ |
| gcongr |
| have H := hfree L hL c A ht ht2 hη hη0 hε E hE hprob (X i) |
| calc |
| _ ≤ ENNReal.ofReal (Real.exp (t*Cdel*(n:ℝ)*r/L.val^3))* |
| ENNReal.ofReal (Cfree*η*ε^(1/3:ℝ)*Real.exp (-spatialSiteDistance L B (X i))) := by gcongr |
| _ = _ := by rw [←ENNReal.ofReal_mul (Real.exp_nonneg _),hDexp]; congr 1; dsimp [z]; ring |
| _ ≤ ENNReal.ofReal (Real.exp (Csp*z*D*(B.card:ℝ))) := |
| hsp n K L hL r D hDeq hD hαr hr hsize hs g hg B z hz hz1 |
| _ ≤ _ := by |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| have hc : (B.card:ℝ)≤(A.card:ℝ) := by exact_mod_cast Finset.card_image_le |
| exact mul_le_mul_of_nonneg_left hc (by positivity) |
| · have he : A=∅ := Finset.not_nonempty_iff_eq_empty.mp hA |
| subst A |
| simp [defectiveTotalVisits,pathVisitedSites] |
|
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| end HardSphere.ShiftVisits |
|
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| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.ShiftVisits |
| open HardSphere.Wiener HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital HardSphere.FreeVisits |
|
|
| def visitorCount {n K : ℕ} (L : Length) (c : Torus L) (E : Set (ConfigurationPath 1 L)) |
| (v : DiscreteTorus K) (ω : ConfigurationPath n L) : ℕ := by |
| classical |
| exact (Finset.univ.filter (fun i : Fin n ↦ onePathAt i ω∈E ∧ pathVisitsBall L 12 (center L c v) i ω)).card |
|
|
| lemma visitorCount_sum {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| (E : Set (ConfigurationPath 1 L)) (ω : ConfigurationPath n L) : |
| (∑v∈A,visitorCount L c E v ω)=defectiveTotalVisits L c A E ω := by |
| classical |
| simp only [visitorCount,Finset.card_filter] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases hi : onePathAt i ω∈E |
| · simp only [hi,true_and,pathVisitedSites,ite_eq_left,Finset.card_filter] |
| · simp only [hi,false_and,ite_false,Finset.sum_const_zero,pathVisitedSites] |
|
|
| lemma visitorCount_measurable {n K : ℕ} (L : Length) (c : Torus L) {E : Set (ConfigurationPath 1 L)} |
| (hE : MeasurableSet E) (v : DiscreteTorus K) : Measurable (visitorCount (n:=n) L c E v) := by |
| classical |
| have he : visitorCount (n:=n) L c E v=(fun ω ↦ ∑i : Fin n, if onePathAt i ω∈E ∧ pathVisitsBall L 12 (center L c v) i ω then (1:ℕ) else 0) := by |
| funext ω |
| simp only [visitorCount,Finset.card_filter] |
| rw [he] |
| apply Finset.measurable_fun_sum |
| intro i _ |
| have hi : MeasurableSet {ω : ConfigurationPath n L | onePathAt i ω∈E} := hE.preimage (onePathAt_continuous i).measurable |
| exact measurable_const.ite (hi.inter (pathVisitsBall_isClosed L 12 (center L c v) i).measurableSet) measurable_const |
|
|
| lemma defectiveTotalVisits_measurable {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| {E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E) : Measurable (defectiveTotalVisits (n:=n) L c A E) := by |
| have he : defectiveTotalVisits (n:=n) L c A E=fun ω ↦ ∑v∈A,visitorCount L c E v ω := by |
| funext ω; exact (visitorCount_sum L c A E ω).symm |
| rw [he] |
| exact Finset.measurable_fun_sum _ (fun v _ ↦ visitorCount_measurable L c hE v) |
|
|
| lemma joint_visitor_sum {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| (E : Set (ConfigurationPath 1 L)) {H : ℝ} {ω : ConfigurationPath n L} |
| (h : ∀v∈A,H≤(visitorCount L c E v ω:ℝ)) : |
| H*(A.card:ℝ)≤(defectiveTotalVisits L c A E ω:ℝ) := by |
| rw [←visitorCount_sum L c A E ω,Nat.cast_sum] |
| have hs := Finset.sum_le_sum h |
| simpa only [Finset.sum_const,nsmul_eq_mul,mul_comm] using hs |
|
|
| lemma joint_visitors_markov {n K : ℕ} (L : Length) (c : Torus L) (A : Finset (DiscreteTorus K)) |
| {E : Set (ConfigurationPath 1 L)} (hE : MeasurableSet E) |
| (μ : Measure (ConfigurationPath n L)) {η H Q : ℝ} (hη : 0≤η) |
| (hmgf : (∫⁻ω,ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ))) ∂μ)≤ |
| ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))) : |
| μ {ω | ∀v∈A,H≤(visitorCount L c E v ω:ℝ)}≤ENNReal.ofReal (Real.exp ((Q-η*H)*(A.card:ℝ))) := by |
| let F := fun ω : ConfigurationPath n L ↦ ENNReal.ofReal (Real.exp (η*(defectiveTotalVisits L c A E ω:ℝ))) |
| have hc : Measurable (fun k : ℕ ↦ (k:ℝ)) := measurable_of_countable _ |
| have hF : Measurable F := ((measurable_const.mul (hc.comp (defectiveTotalVisits_measurable L c A hE))).exp).ennreal_ofReal |
| have hp : ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr (Real.exp_pos _)) |
| have hm := meas_ge_le_lintegral_div (μ:=μ) hF.aemeasurable hp ENNReal.ofReal_ne_top |
| calc |
| _ ≤ μ {ω | ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ))))≤F ω} := by |
| apply measure_mono |
| intro ω hω |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonneg_left (joint_visitor_sum L c A E hω) hη |
| _ ≤ _ := hm |
| _ ≤ ENNReal.ofReal (Real.exp (Q*(A.card:ℝ)))/ENNReal.ofReal (Real.exp (η*(H*(A.card:ℝ)))) := ENNReal.div_le_div_right hmgf _ |
| _ = _ := by |
| rw [←ENNReal.ofReal_div_of_pos (Real.exp_pos _),←Real.exp_sub] |
| congr 2 |
| ring |
|
|
| end HardSphere.ShiftVisits |
|
|
| end |
| end |
|
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| end |
|
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| end OAI |
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