| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct |
| import OAI.Probability.CriticalPercolation.Gluing.RelayMoves |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma vertexCollar_mono {S T : Set V} (hST : S⊆T) (R : ℕ) : |
| G.vertexCollar S R⊆G.vertexCollar T R := by |
| intro y hy |
| obtain ⟨v,hv,hy⟩ := Set.mem_iUnion₂.mp hy |
| exact Set.mem_iUnion₂.mpr ⟨v,hST hv,hy⟩ |
|
|
| lemma initial_mark_subset_relay (t z : V → ℝ) (σ e ρ : ℝ) (hρ : ρ ≤ e) : |
| relayMark t z 0 σ ρ⊆relayRegion t z σ e 0 := by |
| intro v hv |
| obtain ⟨ht₀,ht₁,hz⟩ := hv |
| change |t v-(0:ℕ)| ≤ ((0:ℕ)+1)*e ∧ |z v-σ*(0:ℕ)/4| ≤ Relay.width 0 |
| norm_num [Relay.width] at ht₀ ht₁ hz ⊢ |
| exact ⟨by rw [abs_of_nonneg ht₀]; linarith,hz⟩ |
|
|
| lemma relay_coordinate_scale (L R m : ℕ) (e ρ : ℝ) (he : 0<e) (hρ : 0 ≤ ρ) |
| (heL : (L+3)*e<1/100) (hr : (R+m+1)*ρ<e/10) : |
| ρ ≤ e ∧ (L+1)*e+R*ρ<1 ∧ R*ρ+e/4 ≤ 1/4 ∧ R*ρ+e/4 ≤ e ∧ |
| (L+1)*e+(R+m)*ρ+e/4<1/2 := by |
| have hR : 0 ≤ (R:ℝ)*ρ := mul_nonneg (Nat.cast_nonneg _) hρ |
| have hm : 0 ≤ (m:ℝ)*ρ := mul_nonneg (Nat.cast_nonneg _) hρ |
| have hL : 0 ≤ (L:ℝ)*e := mul_nonneg (Nat.cast_nonneg _) he.le |
| constructor |
| · nlinarith |
| constructor |
| · nlinarith |
| constructor |
| · nlinarith |
| constructor <;> nlinarith |
|
|
| lemma relay_collar_in_domain (t z : V → ℝ) (σ e ρ J : ℝ) (L R i : ℕ) |
| (hσ : |σ|=1) (he : 0 ≤ e) (hρ : 0 ≤ ρ) (hJ : 26200<J) |
| (hi : i ≤ L) (hsmall : (L+1)*e+R*ρ<1) |
| (ht : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v| ≤ ρ) |
| (hz : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v| ≤ ρ) : |
| G.vertexCollar (relayRegion t z σ e i) R⊆relayDomain t z J (L+1/2) σ := by |
| intro y hy |
| have hy' := G.relay_collar_bounds t z σ e ρ hρ ht hz R i hy |
| apply Relay.move_path_contained J L i hi hJ (t y,z y) (t y,z y) |
| (R*ρ) e σ hσ he hsmall hy'.1 hy'.2 <;> norm_num |
|
|
| lemma forceOutside_induced (S : Set V) (ω : Set E) : |
| BernoulliProduct.forceOn (G.inducedEdges S)ᶜ ∅ ω=ω∩G.inducedEdges S := by |
| ext b |
| by_cases hb : b∈G.inducedEdges S <;> simp [hb] |
|
|
| def corridorRelay (t z : V → ℝ) (σ e ρ : ℝ) (L i : ℕ) : Set V := |
| if i=0 then relayMark t z 0 σ ρ |
| else if i ≤ L then relayRegion t z σ e i else relayMark t z (L+1/2) σ ρ |
|
|
| lemma corridorRelay_subset (t z : V → ℝ) (σ e ρ : ℝ) (hρe : ρ ≤ e) |
| (L i : ℕ) (hi : i ≤ L) : corridorRelay t z σ e ρ L i⊆relayRegion t z σ e i := by |
| by_cases h0 : i=0 |
| · subst i |
| simpa [corridorRelay] using initial_mark_subset_relay t z σ e ρ hρe |
| · simp only [corridorRelay,ite_eq_right h0,ite_eq_left hi,Subset.rfl] |
|
|
| theorem coordinate_moves_relay_estimates (t z : V → ℝ) (σ e ρ J : ℝ) (L R m : ℕ) |
| (hσ : |σ|=1) (he : 0<e) (hρ : 0 ≤ ρ) (hJ : 26200<J) (hL : 416 ≤ L) |
| (heL : (L+3)*e<1/100) (hr : (R+m+1)*ρ<e/10) |
| (ht : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v| ≤ ρ) |
| (hz : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v| ≤ ρ) |
| (p : unitInterval) (η : ℝ) |
| (hmove : ∀ y s, |s|=1 → 1-η ≤ (BernoulliProduct.law (fun _ : E => p)).real |
| (G.coordinateMove t z m e y s)) : |
| ∀ i ≤ L, |
| G.vertexCollar (corridorRelay t z σ e ρ L i) R⊆relayDomain t z J (L+1/2) σ ∧ |
| ∀ y∈G.vertexCollar (corridorRelay t z σ e ρ L i) R, |
| 1-η ≤ (BernoulliProduct.law (fun _ : E => p)).real |
| ((BernoulliProduct.forceOn (G.inducedEdges (relayDomain t z J (L+1/2) σ))ᶜ ∅) ⁻¹' |
| G.seedHitEvent (G.vertexBall y m) (corridorRelay t z σ e ρ L (i+1))) := by |
| obtain ⟨hρe,hsmall,hheight,htime,hlast⟩ := relay_coordinate_scale L R m e ρ he hρ heL hr |
| intro i hi |
| have hsub := G.vertexCollar_mono (corridorRelay_subset t z σ e ρ hρe L i hi) R |
| refine ⟨hsub.trans (G.relay_collar_in_domain t z σ e ρ J L R i hσ he.le hρ hJ hi hsmall ht hz),?_⟩ |
| intro y hy |
| apply (hmove y (Relay.selectedSign (z y-σ*(i:ℝ)/4)) (Relay.selectedSign_abs _)).trans |
| apply measureReal_mono (h₂ := measure_ne_top _ _) |
| intro ω hω |
| rw [Set.mem_preimage,G.forceOutside_induced] |
| by_cases hiL : i<L |
| · have hh := G.full_relay_move t z σ e ρ J L R m i hσ he.le hρ hJ hiL hsmall |
| hheight htime ht hz (hsub hy) hω |
| change ω∩G.inducedEdges (relayDomain t z J (L+1/2) σ)∈ |
| G.seedHitEvent (G.vertexBall y m) (relayRegion t z σ e (i+1)) at hh |
| simpa only [corridorRelay,Nat.add_one_ne_zero,ite_false,ite_eq_left (by omega : i+1 ≤ L)] using hh |
| · have hiEq : i=L := by omega |
| subst i |
| have hh := G.last_relay_move t z σ e ρ J L R m hσ he.le hρ hJ hL hlast ht hz |
| (hsub hy) _ hω |
| change ω∩G.inducedEdges (relayDomain t z J (L+1/2) σ)∈ |
| G.seedHitEvent (G.vertexBall y m) (relayMark t z (L+1/2) σ ρ) at hh |
| simpa only [corridorRelay,Nat.add_one_ne_zero,ite_false,ite_eq_right (by omega : ¬L+1 ≤ L)] using hh |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
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|
