| import OAI.Probability.FirstPassage.FiniteEdgeTilt |
| import OAI.Probability.FirstPassage.TiltRegionCount |
| import OAI.Probability.FirstPassage.TiltPath |
| import OAI.Probability.FirstPassage.RareFast |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Set Finset MeasureTheory ProbabilityTheory Filter |
| open scoped Topology |
|
|
| noncomputable def crossingEvent (b m K s : ℝ) (n : ℤ) (t : Vertex) : Set (Edge → ℝ) := |
| strictLocalizedEvent (rareFastVertices b 1 8 s n t) |
| (fun p ds => p.1=t.1 ∧ (p+displacement ds).1=t.1+n) |
| (fun _ _ => m*n+K*s) |
|
|
| theorem measurableSet_crossingEvent (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| MeasurableSet (crossingEvent b m K s n t) := measurableSet_strictLocalizedEvent _ _ _ |
|
|
| theorem crossingEvent_depends (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| DependsOnEdges (crossingEvent b m K s n t) (internalEdges (rareFastVertices b 1 8 s n t)) := |
| strictLocalizedEvent_depends _ _ _ |
|
|
| theorem crossingEvent_tilt_implication (b m a K lam s : ℝ) (n : ℤ) |
| (hK : 0 ≤ K) (hlam : 0 ≤ lam) (hs : 0 < s) (hn : 0 < n) |
| (hsmall : lam*s < n) (hgain : K+16*|a|+1 < lam*m) |
| (w : Edge → ℝ) (hw : w ∈ crossingEvent b m K s n 0) : |
| scaleFiniteEdges (internalEdges (rareFastVertices b 1 8 s n 0)) (1-lam*s/n) w ∈ |
| rareFastEvent b m a 1 8 1 s n 0 := by |
| rw [scaleFiniteEdges_event_iff _ _ _ (rareFastEvent_depends b m a 1 8 1 s n 0)] |
| obtain ⟨p,ds,⟨hp,hq⟩,hin,hcost⟩ := hw |
| have hx : ((displacement ds).1 : ℝ)=(n : ℝ) := by |
| have he : (displacement ds).1=n := by simpa [Prod.fst_add,hp] using hq |
| exact_mod_cast he |
| have hstart : p ∈ rareFastRegion b 1 8 s n 0 := by |
| have hh := hin 0 (by omega) |
| simpa using hh |
| have hend : p+displacement ds ∈ rareFastRegion b 1 8 s n 0 := by |
| exact (mem_rareFastVertices ..).mp (pathIn_endpoint _ _ _ hin) |
| have hr : |transverse b (displacement ds)| ≤ 16*s := by |
| have ht := abs_sub (transverse b (p+displacement ds)) (transverse b p) |
| simp only [transverse_add,add_sub_cancel_left] at ht |
| have h1 := hstart.2.2 |
| have h2 := hend.2.2 |
| simp only [sub_zero,transverse_add] at h1 h2 |
| linarith |
| have hd := tilt_crossing_defect w b m a K lam s n hK hlam hs |
| (by exact_mod_cast hn) hsmall hgain p ds hx hr hcost |
| apply (mem_rareFastEvent ..).mpr |
| refine ⟨p,ds,hp,hq,?_,?_⟩ |
| · intro i hi |
| exact (mem_rareFastVertices ..).mp (hin i hi) |
| · simpa using hd |
|
|
| theorem source_tilt_exponent_bound (N : ℕ) (lam s n : ℝ) |
| (_hs : 0 ≤ s) (hn : 0 < n) (hN : (N : ℝ) ≤ 144*n*s) (hn3 : s^3 ≤ n) : |
| 2*N*(lam*s/n)^2 ≤ 288*lam^2 := by |
| calc |
| _ ≤ (2*(144*n*s))*(lam*s/n)^2 := by gcongr |
| _ = (288*lam^2)*(s^3/n) := by field_simp; ring |
| _ ≤ (288*lam^2)*1 := mul_le_mul_of_nonneg_left ((div_le_one hn).mpr hn3) (by positivity) |
| _ = _ := mul_one _ |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
|
|
| theorem exponential_crossing_tilt_bound (h : ExponentialEnvironment P τ) |
| (b m a K lam s : ℝ) (n : ℤ) (hK : 0 ≤ K) (hlam : 0 ≤ lam) |
| (hs : 1 ≤ s) (hn : 1 ≤ n) (hn3 : s^3 ≤ n) |
| (hsmall : lam*s/n ≤ 1/2) (hgain : K+16*|a|+1 < lam*m) : |
| P.real {ω | (fun e => τ e ω) ∈ crossingEvent b m K s n 0} ^ 2 ≤ |
| Real.exp (288*lam^2) * |
| P.real {ω | (fun e => τ e ω) ∈ rareFastEvent b m a 1 8 1 s n 0} := by |
| let _ := h.probability |
| let S := internalEdges (rareFastVertices b 1 8 s n 0) |
| let F := rareFastEvent b m a 1 8 1 s n 0 |
| have hn0 : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega) |
| have hδ : 0 ≤ lam*s/n := by positivity |
| have hlt : lam*s < n := by |
| have hh := (div_le_iff₀ hn0).mp hsmall |
| linarith |
| have himp : {ω | (fun e => τ e ω) ∈ crossingEvent b m K s n 0} ⊆ |
| {ω | scaleFiniteEdges S (1-lam*s/n) (fun e => τ e ω) ∈ F} := by |
| intro ω hω |
| exact crossingEvent_tilt_implication b m a K lam s n hK hlam (by linarith) (by omega) hlt hgain _ hω |
| have hprob := measureReal_mono (μ := P) himp |
| have hcomp := finite_edge_tilt_probability_sq_exp_bound h S (lam*s/n) hδ hsmall F |
| (measurableSet_rareFastEvent ..) (rareFastEvent_depends b m a 1 8 1 s n 0) |
| have hexp : Real.exp (2*S.card*(lam*s/n)^2) ≤ Real.exp (288*lam^2) := by |
| apply Real.exp_le_exp.mpr |
| exact source_tilt_exponent_bound S.card lam s n (by linarith) hn0 |
| (rareFast_tilt_edges_card b s n hs hn) hn3 |
| exact (pow_le_pow_left₀ measureReal_nonneg hprob 2).trans |
| (hcomp.trans (mul_le_mul_of_nonneg_right hexp measureReal_nonneg)) |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
