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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
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