| import Mathlib |
| import OAI.Analysis.HardSphere.PathBounds.VisitCounts |
| import OAI.Analysis.HardSphere.Cutoffs.StandardRegular |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.FreeVisits |
| open Wiener PathCutoff HardSphere.Lattice |
|
|
| theorem ground_regularVisitors_tail (η : ℝ) (hη : 0 < η) : |
| ∃ αstar : ℝ, 0 < αstar ∧ ∀ αmax C a b : ℝ, |
| 0 < αmax → 0 < C → 0 < a → 0 < b → |
| ∃ D₀ : ℝ, 1 ≤ D₀ ∧ ∀ (n K : ℕ) [NeZero K] (L : Length), L.val=(K:ℝ) → |
| ∀ r D : ℝ, D=(n:ℝ)/L.val^3 → D₀ ≤ D → αstar ≤ D*r → D*r ≤ αmax → 0 < r → |
| (n:ℝ)*(2*Real.pi*r)^3 ≤ L.val^3/2 → |
| ∀ (g : Jet n L), IsGroundVector n L r g → ∀ (A : Finset (DiscreteTorus K)) |
| (m : ℕ) (τ : ℝ), 0 < m → 0 < τ → τ ≤ r^2 → |
| a*D^2 ≤ (m:ℝ) → (m:ℝ) ≤ C*D^2 → (m:ℝ)*τ ≤ 2 → |
| ∀ (hD : 0 < D) (hhT : D⁻¹ < (m:ℝ)*τ), |
| groundPathLaw n L r ((m:ℝ)*τ) g {ω | ∀v∈A,1 ≤ |
| (visitorCount L {p | ¬Regular m r (D^(1/12:ℝ)) (meanCost+1) |
| (splitTime D⁻¹ ((m:ℝ)*τ-D⁻¹) (inv_pos.mpr hD) (sub_pos.mpr hhT)) |
| (b*D^(-(1/3:ℝ))/8) p} v ω:ℝ)} ≤ |
| ENNReal.ofReal (Real.exp (-(η/2)*(A.card:ℝ))) := by |
| obtain ⟨c,M,hc,hM,htail⟩ := wiener_norm_gaussian_tail |
| obtain ⟨Cd,Cf,Cs,α,hCd,hCf,hCs,hα,hm⟩ := interacting_visit_mgf (η₀:=η) hη.le |
| refine ⟨α,hα,?_⟩ |
| intro αmax C a b hαmax hC ha hb |
| let ε := fun D : ℝ ↦ scalarRegularBound C (3*M.toReal) c a b D |
| let B := Real.exp (2*Cd*αmax)*Cf*η |
| have hB : 0 < B := by dsimp [B]; positivity |
| let δ := min 1 (η/(2*Cs)) |
| have hδ : 0 < δ := lt_min (by norm_num) (by positivity) |
| have hlim : Tendsto (fun D : ℝ ↦ B*(D*(ε D)^(1/3:ℝ))) atTop (𝓝 0) := by |
| simpa using (regular_root_decay hC.le (by positivity : 0 ≤ 3*M.toReal) hc ha hb).const_mul B |
| obtain ⟨D₀,hsmall⟩ := (eventually_atTop.1 |
| ((eventually_ge_atTop (1:ℝ)).and (hlim.eventually (gt_mem_nhds hδ)))) |
| refine ⟨max 1 D₀,le_max_left _ _,?_⟩ |
| intro n K inst L hL r D hDeq hDD hαr hαup hr hs g hg A m τ hmpos hτ hτr hmlo hmhi ht2 hD hhT |
| have hD1 : 1 ≤ D := (le_max_left _ _).trans hDD |
| have hsmallD : B*(D*(ε D)^(1/3:ℝ)) ≤ δ := ((hsmall D ((le_max_right _ _).trans hDD)).2).le |
| have hε : 0 ≤ ε D := scalarRegularBound_nonneg hC.le (by positivity) c a b D |
| have ht : 0 < (m:ℝ)*τ := mul_pos (Nat.cast_pos.mpr hmpos) hτ |
| have hxp : ((m:ℝ)*τ)*Cd*D*r ≤ 2*Cd*αmax := by |
| calc |
| _ = (((m:ℝ)*τ)*Cd)*(D*r) := by ring |
| _ ≤ (((m:ℝ)*τ)*Cd)*αmax := mul_le_mul_of_nonneg_left hαup (by positivity) |
| _ ≤ (2*Cd)*αmax := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right ht2 hCd.le) hαmax.le |
| let z := Real.exp (((m:ℝ)*τ)*Cd*D*r)*(Cf*η*(ε D)^(1/3:ℝ)) |
| have hz : 0 ≤ z := by dsimp [z]; positivity |
| have hzD : z*D ≤ δ := by |
| apply le_trans _ hsmallD |
| calc |
| _ ≤ Real.exp (2*Cd*αmax)*(Cf*η*(ε D)^(1/3:ℝ))*D := by |
| dsimp [z] |
| exact mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hxp) (by positivity)) hD.le |
| _ = B*(D*(ε D)^(1/3:ℝ)) := by dsimp [B]; ring |
| have hz1 : z ≤ 1 := (by nlinarith : z ≤ z*D).trans (hzD.trans (min_le_left _ _)) |
| let E : Set (ConfigurationPath 1 L) := {p | ¬Regular m r (D^(1/12:ℝ)) (meanCost+1) |
| (splitTime D⁻¹ ((m:ℝ)*τ-D⁻¹) (inv_pos.mpr hD) (sub_pos.mpr hhT)) (b*D^(-(1/3:ℝ))/8) p} |
| have hE : MeasurableSet E := (regular_measurable _ _ _ _ _ _).compl |
| have hrev (p : ConfigurationPath 1 L) : reverseConfigurationPath 1 L p∈E ↔ p∈E := |
| (regular_reverse _ _ _ _ _ _ p).not |
| have hp (y : Torus L) : coordinateWiener 1 {w | freeTrajectory 1 L ((m:ℝ)*τ) w (oneConfiguration y)∈E} ≤ ENNReal.ofReal (ε D) := |
| free_regular_standard_tail hM.ne htail hr hτ hτr hc.le hD1 hC.le ha.le hb.le m hmpos hmlo hmhi hhT (oneConfiguration y) |
| have hh := hm n K L hL r D hDeq hD1 hαr hr hs g hg A ht ht2 hη.le (le_refl η) hε E hE hrev hp hz1 |
| have hj := joint_visitors_markov L A hE (groundPathLaw n L r ((m:ℝ)*τ) g) hη.le hh (H:=1) |
| apply hj.trans |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg A.card) |
| have hδη : Cs*δ ≤ η/2 := by |
| have hh := (le_div_iff₀ (by positivity : 0 < (2:ℝ)*Cs)).mp (min_le_right 1 (η/(2*Cs))) |
| dsimp [δ] |
| linarith |
| have hcost := (mul_le_mul_of_nonneg_left hzD hCs.le).trans hδη |
| change Cs*z*D-η*1 ≤ -(η/2) |
| nlinarith |
|
|
| end HardSphere.FreeVisits |
|
|
| end |
| end |
|
|
| end |
|
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| end OAI |
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