| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.BridgeStrongRegular |
| import OAI.Analysis.HardSphere.Bridges.BridgeBlocks |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology |
| namespace HardSphere |
| open Wiener Bridge GaussianPacking |
|
|
| def labelStartLift {N : ℕ} {L : Length} (e : BridgeEndpoints N L) (i : Fin N) : Point := |
| representative N L e.2 i |
|
|
| def labelEndLift {N : ℕ} {L : Length} (t : ℝ) (e : BridgeEndpoints N L) (i : Fin N) : Point := |
| fun j ↦ labelStartLift e i j+Real.sqrt (2*t)*e.1 (i,j) |
|
|
| lemma labelBridgePath_lift {N : ℕ} {L : Length} (t : ℝ) |
| (e : BridgeEndpoints N L) (β : Fin N→Bridge.Noise) (i : Fin N) : |
| onePathAt i (labelBridgePath N L t (e,β))= |
| projectCurve L (leg t (labelStartLift e i) (labelEndLift t e i) (β i)) := by |
| ext s k j |
| have hp := congrFun (congrFun (project_representative N L e.2) i) j |
| change e.2 i j+((Real.sqrt (2*t)*(e.1 (i,j)*(s:ℝ)+β i j s):ℝ):AddCircle L.val)= |
| (((1-(s:ℝ))*labelStartLift e i j+(s:ℝ)*labelEndLift t e i j+Real.sqrt (2*t)*β i j s:ℝ):AddCircle L.val) |
| change ((labelStartLift e i j:ℝ):AddCircle L.val)=e.2 i j at hp |
| rw [←hp,←QuotientAddGroup.mk_add] |
| congr 1 |
| unfold labelEndLift |
| ring |
|
|
| lemma labelBridgePath_one_law {N : ℕ} {L : Length} (t : ℝ) |
| (e : BridgeEndpoints N L) (i : Fin N) : |
| (Measure.pi fun _ : Fin N ↦ Bridge.law).map |
| (fun β ↦ onePathAt i (labelBridgePath N L t (e,β)))= |
| (Bridge.law.map (leg t (labelStartLift e i) (labelEndLift t e i))).map (projectCurve L) := by |
| simp_rw [labelBridgePath_lift] |
| have h := congrArg (fun μ : Measure Bridge.Noise ↦ μ.map |
| (projectCurve L ∘ leg t (labelStartLift e i) (labelEndLift t e i))) |
| (measurePreserving_eval (fun _ : Fin N ↦ Bridge.law) i).map_eq |
| rw [Measure.map_map ((projectCurve_continuous L).measurable.comp (leg_continuous t _ _).measurable) (measurable_pi_apply i)] at h |
| rw [Measure.map_map (projectCurve_continuous L).measurable (leg_continuous _ _ _).measurable] |
| exact h |
|
|
| end HardSphere |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
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|
