| import OAI.Probability.FirstPassage.EndpointNet |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set Filter Finset MeasureTheory ProbabilityTheory |
| open scoped Topology |
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| def BallTravel (μ : Plane → ℝ) (D η δ s : ℝ) (w : Edge → ℝ) : Prop := |
| ∀ p q : Vertex, ‖latticeEmbed p‖ ≤ D*s → ‖latticeEmbed q‖ ≤ D*s → |
| LocalTravel μ η δ s p q w |
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| theorem ballTravel_of_net (μ : Plane → ℝ) (w : Edge → ℝ) |
| (D γ β η δ κ : ℝ) (hβ : 0 ≤ β) |
| (hlip : ∀ v u, |μ v-μ u| ≤ β*‖v-u‖) |
| (S : Finset {v : Plane // ‖v‖ ≤ D}) |
| (hS : ∀ v : Plane, ‖v‖ ≤ D → ∃ u ∈ S, ‖v-u.val‖ < γ) |
| (C n : ℕ) (hn : 0 < n) (hbox : D*n+2 ≤ (C*n : ℕ)) |
| (hcost : η/4*n+2*(8*(γ*n+2)+κ*(C*n : ℕ))+2*β*(γ*n+2) ≤ η*n) |
| (htube : max (3*(γ*n+2)) (δ/2*n+(γ*n+2)) ≤ δ*n) |
| (hc : ∀ p ∈ latticeBox (C*n), ∀ q ∈ latticeBox (C*n), |
| pathCost w p (coordinatePath p q) < 4*(latticeDistance p q : ℝ)+κ*(C*n : ℕ)) |
| (hm : ∀ u ∈ S, ∀ v ∈ S, LocalTravel μ (η/4) (δ/2) n |
| (latticeFloor ((n : ℝ) • u.val)) (latticeFloor ((n : ℝ) • v.val)) w) : |
| BallTravel μ D η δ n w := by |
| intro p q hp hq |
| obtain ⟨u,hu,hpu⟩ := scaled_net_approximates D γ S hS n hn p hp |
| obtain ⟨v,hv,hqv⟩ := scaled_net_approximates D γ S hS n hn q hq |
| let p' := latticeFloor ((n : ℝ) • u.val) |
| let q' := latticeFloor ((n : ℝ) • v.val) |
| have hpbox : p ∈ latticeBox (C*n) := lattice_mem_box_of_norm p (C*n) (hp.trans (by linarith)) |
| have hqbox : q ∈ latticeBox (C*n) := lattice_mem_box_of_norm q (C*n) (hq.trans (by linarith)) |
| have hp'box : p' ∈ latticeBox (C*n) := lattice_mem_box_of_norm p' (C*n) |
| ((scaled_net_norm D u n).trans hbox) |
| have hq'box : q' ∈ latticeBox (C*n) := lattice_mem_box_of_norm q' (C*n) |
| ((scaled_net_norm D v n).trans hbox) |
| have hleft : pathCost w p (coordinatePath p p') ≤ 8*(γ*n+2)+κ*(C*n : ℕ) := by |
| have hd := latticeDistance_le_norm p p' |
| rw [← latticeEmbed_sub] at hd |
| have hh := hc p hpbox p' hp'box |
| dsimp [p'] at * |
| linarith |
| have hright : pathCost w q' (coordinatePath q' q) ≤ 8*(γ*n+2)+κ*(C*n : ℕ) := by |
| have hd := latticeDistance_le_norm q' q |
| rw [norm_sub_rev,← latticeEmbed_sub] at hd |
| have hh := hc q' hq'box q hqbox |
| dsimp [q'] at * |
| linarith |
| obtain ⟨ds,hend,hc',ht'⟩ := localTravel_attach μ w β hβ hlip p q p' q' |
| (γ*n+2) (8*(γ*n+2)+κ*(C*n : ℕ)) (η/4) (δ/2) n hpu hqv hleft hright (hm u hu v hv) |
| refine ⟨ds,hend,hc'.trans (by linarith),?_⟩ |
| intro i hi |
| obtain ⟨t,ht0,ht1,hbound⟩ := ht' i hi |
| exact ⟨t,ht0,ht1,hbound.trans htube⟩ |
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| theorem ballTravel_parameters (D β η δ : ℝ) |
| (hD : 0 < D) (hβ : 0 ≤ β) (hη : 0 < η) (hδ : 0 < δ) : |
| ∃ γ κ : ℝ, ∃ C : ℕ, 0 < γ ∧ 0 < κ ∧ 0 < C ∧ |
| ∀ᶠ n : ℕ in atTop, |
| 0 < n ∧ D*n+2 ≤ (C*n : ℕ) ∧ |
| η/4*n+2*(8*(γ*n+2)+κ*(C*n : ℕ))+2*β*(γ*n+2) ≤ η*n ∧ |
| max (3*(γ*n+2)) (δ/2*n+(γ*n+2)) ≤ δ*n := by |
| obtain ⟨C,hC⟩ := exists_nat_gt (D+2) |
| have hc : (0 : ℝ) < C := by linarith |
| have hc' : 0 < C := by exact_mod_cast hc |
| let γ := min (η/(8*(16+2*β))) (δ/12) |
| let κ := η/(16*(C : ℝ)) |
| have hγ : 0 < γ := lt_min (div_pos hη (by positivity)) (by positivity) |
| have hκ : 0 < κ := div_pos hη (by positivity) |
| have hgη : (16+2*β)*γ ≤ η/8 := by |
| have ht := (le_div_iff₀ (by positivity : 0 < 8*(16+2*β))).mp (min_le_left _ _ : γ ≤ _) |
| nlinarith |
| have hgδ : γ ≤ δ/12 := min_le_right _ _ |
| have hk : κ*(C : ℝ)=η/16 := by dsimp [κ]; field_simp |
| refine ⟨γ,κ,C,hγ,hκ,hc',?_⟩ |
| have hlim := (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop) |
| filter_upwards [eventually_ge_atTop 1, |
| hlim.eventually (eventually_ge_atTop (4*(32+4*β)/η)), |
| hlim.eventually (eventually_ge_atTop (24/δ))] with n hn hnη hnδ |
| have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn |
| have hn0 : (0 : ℝ) ≤ n := by positivity |
| have hhη := (div_le_iff₀ hη).mp hnη |
| have hhδ := (div_le_iff₀ hδ).mp hnδ |
| have hge := mul_le_mul_of_nonneg_right hgη hn0 |
| have hgd := mul_le_mul_of_nonneg_right hgδ hn0 |
| have hcb := mul_le_mul_of_nonneg_right hC.le hn0 |
| simp only [Nat.cast_mul] |
| refine ⟨by omega,by nlinarith,?_,?_⟩ |
| · have he : κ*((C : ℝ)*n)=(η/16)*n := by rw [← mul_assoc,hk] |
| rw [he] |
| nlinarith |
| · apply max_le <;> nlinarith |
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| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
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| theorem badConnections_tendsto_zero (h : ExponentialEnvironment P τ) |
| (κ : ℝ) (hκ : 0 < κ) : |
| Tendsto (fun n => P.real (badConnections τ κ n)) atTop (𝓝 0) := by |
| apply squeeze_zero (fun _ => measureReal_nonneg) (badConnections_probability h κ) |
| exact (summable_box_connection_bound hκ).tendsto_atTop_zero |
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| theorem exponential_ball_travel_nat (h : ExponentialEnvironment P τ) |
| (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ) |
| (D η δ : ℝ) (hD : 0 < D) (hη : 0 < η) (hδ : 0 < δ) : |
| Tendsto (fun n : ℕ => P.real {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)}) |
| atTop (𝓝 0) := by |
| classical |
| let _ := h.probability |
| obtain ⟨β,hβ,hlip⟩ := timeConstantNorm_lipschitz_bound μ hμ |
| obtain ⟨γ,κ,C,hγ,hκ,hC,hparams⟩ := ballTravel_parameters D β η δ hD hβ.le hη hδ |
| obtain ⟨S,hS⟩ := finite_ball_net D γ hγ |
| let A (n : ℕ) := ⋃ uv ∈ S ×ˢ S, {ω | ¬LocalTravel μ (η/4) (δ/2) n |
| (latticeFloor ((n : ℝ) • uv.1.val)) (latticeFloor ((n : ℝ) • uv.2.val)) (fun e => τ e ω)} |
| have hA : Tendsto (fun n => P.real (A n)) atTop (𝓝 0) := by |
| apply (tendsto_order.2 ⟨?_,?_⟩) |
| · intro a ha |
| exact Eventually.of_forall (fun _ => ha.trans_le measureReal_nonneg) |
| · intro ε hε |
| have hden : 0 < ((S ×ˢ S).card : ℝ)+1 := by positivity |
| filter_upwards [eventually_ge_atTop 1, |
| exponential_local_travel_nat h μ hμ ht (2*D+4) (η/4) (δ/2) |
| (by linarith) (by linarith) (by linarith) |
| (ε/(((S ×ˢ S).card : ℝ)+1)) (div_pos hε hden)] with n hn hh |
| have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn |
| have hbound : P.real (A n) ≤ ((S ×ˢ S).card : ℝ)*(ε/(((S ×ˢ S).card : ℝ)+1)) := by |
| refine (measureReal_biUnion_finset_le _ _).trans ?_ |
| calc |
| _ ≤ ∑ _uv ∈ S ×ˢ S, ε/(((S ×ˢ S).card : ℝ)+1) := by |
| apply sum_le_sum |
| intro uv huv |
| apply (hh _ _ ?_).le |
| rw [latticeEmbed_sub] |
| exact (norm_sub_le _ _).trans (by |
| have h1 := scaled_net_norm D uv.1 n |
| have h2 := scaled_net_norm D uv.2 n |
| nlinarith) |
| _ = _ := by simp |
| apply hbound.trans_lt |
| have he := (div_pos hε hden) |
| have hid : ((((S ×ˢ S).card : ℝ)+1))*(ε/(((S ×ˢ S).card : ℝ)+1))=ε := by field_simp |
| nlinarith |
| have hCn : Tendsto (fun n : ℕ => C*n) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro b |
| filter_upwards [eventually_ge_atTop b] with n hn |
| exact hn.trans (Nat.le_mul_of_pos_left n hC) |
| have hB := (badConnections_tendsto_zero h κ hκ).comp hCn |
| have hbound : ∀ᶠ n : ℕ in atTop, |
| P.real {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)} ≤ |
| P.real (badConnections τ κ (C*n))+P.real (A n) := by |
| filter_upwards [hparams] with n hn |
| have hsub : {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)} ⊆ |
| badConnections τ κ (C*n) ∪ A n := by |
| intro ω hω |
| by_contra hb |
| have hc : ω ∉ badConnections τ κ (C*n) := fun hc => hb (Or.inl hc) |
| have hm : ω ∉ A n := fun hm => hb (Or.inr hm) |
| apply hω |
| apply ballTravel_of_net μ _ D γ β η δ κ hβ.le hlip S hS C n |
| hn.1 hn.2.1 hn.2.2.1 hn.2.2.2 |
| · intro p hp q hq |
| by_contra hh |
| exact hc (mem_iUnion₂.mpr ⟨(p,q),mem_product.mpr ⟨hp,hq⟩,le_of_not_gt hh⟩) |
| · intro u hu v hv |
| by_contra hh |
| exact hm (mem_iUnion₂.mpr ⟨(u,v),mem_product.mpr ⟨hu,hv⟩,hh⟩) |
| exact (measureReal_mono hsub).trans (measureReal_union_le _ _) |
| exact squeeze_zero' (Eventually.of_forall (fun _ => measureReal_nonneg)) hbound |
| (by simpa using hB.add hA) |
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| end PlanarFPP |
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| end OAI |
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