| import Mathlib |
| import OAI.Probability.CriticalPercolation.Modular.NormalizedDiscrepancy |
| import OAI.Probability.CriticalPercolation.Modular.DiscrepancyTail |
| import OAI.Probability.CriticalPercolation.Modular.HeavyThinning |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneBernoulli |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology BigOperators |
|
|
| namespace CriticalPercolation |
|
|
| lemma ennreal_tendsto_zero_of_uniform_tail (f g : ℕ → ℝ≥0∞) (c b : ℕ → ℝ≥0∞) |
| (hc : ∀ N, c N≠∞) (hg : Tendsto g atTop (𝓝 0)) (hb : Tendsto b atTop (𝓝 0)) |
| (hle : ∀ N n, f n≤c N*g n+b N) : Tendsto f atTop (𝓝 0) := by |
| apply ENNReal.tendsto_nhds_zero.mpr |
| intro ε hε |
| have hh := ENNReal.half_pos hε.ne' |
| obtain ⟨N,hN⟩ := (ENNReal.tendsto_nhds_zero.mp hb (ε/2) hh).exists |
| have ht : Tendsto (fun n => c N*g n) atTop (𝓝 0) := by |
| simpa only [mul_zero] using ENNReal.Tendsto.const_mul hg (Or.inr (hc N)) |
| filter_upwards [ENNReal.tendsto_nhds_zero.mp ht (ε/2) hh] with n hn |
| exact (hle N n).trans ((add_le_add hn hN).trans_eq (ENNReal.add_halves ε)) |
|
|
| namespace BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) [Countable V] [Countable E] |
|
|
| omit [Countable V] [Countable E] in |
| lemma clusterPeakRatio_ne_zero_of_mass_ne_top (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (x : V) (hm : G.modularMass x (G.cluster ω x)≠∞) : |
| G.clusterPeakRatio ω x≠0 := by |
| obtain ⟨y,hy⟩ := G.clusterPeaks_nonempty_of_mass_ne_top hc hl ω x hm |
| rw [G.clusterPeakRatio_eq_inv_mass hc hl ω x y hm hy] |
| exact ENNReal.inv_ne_zero.mpr (fun he => hm ((G.modularMass_eq_top_change_base hc hl x y _).mpr he)) |
|
|
| omit [Countable V] [Countable E] in |
| lemma normalizedDiscrepancy_le_ratio_add_tail (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) |
| (hK : ∀ v, (G.vertexBall v 1).ncard≤K) (p : unitInterval) (T : ℝ≥0∞) |
| (ω : Set E) (x : V) : |
| G.normalizedAbsoluteDiscrepancy p ω x ≤ |
| (D*(2*K) : ℕ)*T*G.clusterPeakRatio ω x + G.discrepancyVolumeTail p T ω x := by |
| by_cases ht : T≤G.normalizedClusterVolume ω x |
| · rw [discrepancyVolumeTail,ite_eq_left ht] |
| exact le_add_left le_rfl |
| · by_cases hm : G.modularMass x (G.cluster ω x)=∞ |
| · simp only [normalizedAbsoluteDiscrepancy,hm,ENNReal.div_top,zero_le] |
| · have hr := G.clusterPeakRatio_ne_zero_of_mass_ne_top hc hl ω x hm |
| have hr' := ne_top_of_le_ne_top ENNReal.one_ne_top (G.clusterPeakRatio_le_one hc hl ω x) |
| have hv : G.normalizedClusterVolume ω x*G.clusterPeakRatio ω x=1 := by |
| rw [normalizedClusterVolume,ite_eq_right hr,ENNReal.inv_mul_cancel hr hr'] |
| have hone : 1≤T*G.clusterPeakRatio ω x := by |
| rw [←hv] |
| exact mul_le_mul_left (le_of_not_ge ht) _ |
| calc _ ≤ ((D*(2*K) : ℕ) : ℝ≥0∞) := G.normalizedAbsoluteDiscrepancy_le hc hl D K hD hK p p.property ω x |
| _ ≤ (D*(2*K) : ℕ)*T*G.clusterPeakRatio ω x := by |
| simpa only [mul_one,mul_assoc] using mul_le_mul_right hone ((D*(2*K) : ℕ) : ℝ≥0∞) |
| _ ≤ _ := le_add_right le_rfl |
|
|
| lemma expected_heavy_normalizedDiscrepancy_bound |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) |
| (hK : ∀ v, (G.vertexBall v 1).ncard≤K) (p q : unitInterval) (N : ℕ) |
| (x : V) (hx : x∈(G.orbitRepresentatives hq).roots) : |
| (∫⁻ u, if G.heavyFlag p u x then G.normalizedAbsoluteDiscrepancy q (atParameter q u) x else 0 ∂markLaw) ≤ |
| ((D*(2*K) : ℕ)*ENNReal.ofReal ((16:ℝ)^N)) * |
| (∫⁻ u, if G.heavyFlag p u x then G.clusterPeakRatio (atParameter q u) x else 0 ∂markLaw) + |
| (G.orbitRepresentatives hq).roots.card * |
| ENNReal.ofReal (5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1)* |
| (ENNReal.ofReal (1/2:ℝ))^N*2 := by |
| let C : ℝ≥0∞ := (D*(2*K) : ℕ)*ENNReal.ofReal ((16:ℝ)^N) |
| let r := fun u => if G.heavyFlag p u x then G.clusterPeakRatio (atParameter q u) x else 0 |
| have hm : Measurable r := ((G.measurable_clusterPeakRatio x).comp (measurable_atParameter q)).ite |
| (G.measurable_heavyFlag p x).setOf measurable_const |
| calc |
| _ ≤ ∫⁻ u, C*r u+G.discrepancyVolumeTail q (ENNReal.ofReal ((16:ℝ)^N)) (atParameter q u) x ∂markLaw := by |
| apply lintegral_mono |
| intro u |
| by_cases h : G.heavyFlag p u x |
| · simpa only [h,ite_true,r,C] using G.normalizedDiscrepancy_le_ratio_add_tail hc hl D K hD hK q |
| (ENNReal.ofReal ((16:ℝ)^N)) (atParameter q u) x |
| · simp only [h,ite_false,zero_le] |
| _ = C*(∫⁻ u, r u ∂markLaw) + (∫⁻ ω, G.discrepancyVolumeTail q (ENNReal.ofReal ((16:ℝ)^N)) ω x ∂G.law q) := by |
| rw [lintegral_add_left (show Measurable (fun u => C*r u) from measurable_const.mul hm), |
| lintegral_const_mul'' C hm.aemeasurable] |
| congr 1 |
| rw [show G.law q=(markLaw (E := E)).map (atParameter q) from by simpa only [BondGraph.law, BernoulliProduct.law_const] using (markLaw_map_atParameter (E := E) q).symm, |
| lintegral_map (G.measurable_discrepancyVolumeTail q _ x) (measurable_atParameter q)] |
| _ ≤ _ := add_le_add le_rfl (G.expected_discrepancyTail_bound hc hl hq D K hD hK q N x hx) |
|
|
| theorem tendsto_expected_heavy_normalizedDiscrepancy |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) |
| (hK : ∀ v, (G.vertexBall v 1).ncard≤K) |
| (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q) |
| (hqp : ∀ n, q n≤p) (hlim : Tendsto q atTop (𝓝 p)) |
| (x : V) (hx : x∈(G.orbitRepresentatives hq).roots) : |
| Tendsto (fun n => ∫⁻ u, if G.heavyFlag p u x then |
| G.normalizedAbsoluteDiscrepancy (q n) (atParameter (q n) u) x else 0 ∂markLaw) atTop (𝓝 0) := by |
| let r := fun n => ∫⁻ u, if G.heavyFlag p u x then G.clusterPeakRatio (atParameter (q n) u) x else 0 ∂markLaw |
| have hr : Tendsto r atTop (𝓝 0) := by |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds |
| (G.tendsto_sum_expected_heavy_clusterPeakRatio hc hl hq p q hmono hqp hlim) |
| · exact fun _ => zero_le |
| · intro n |
| exact Finset.single_le_sum (f := fun y => ∫⁻ u, |
| if G.heavyFlag p u y then G.clusterPeakRatio (atParameter (q n) u) y else 0 ∂markLaw) |
| (fun _ _ => zero_le) hx |
| apply ennreal_tendsto_zero_of_uniform_tail _ r |
| (fun N => (D*(2*K) : ℕ)*ENNReal.ofReal ((16:ℝ)^N)) |
| (fun N => (G.orbitRepresentatives hq).roots.card * |
| ENNReal.ofReal (5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1)* |
| (ENNReal.ofReal (1/2:ℝ))^N*2) |
| · intro N |
| exact ENNReal.mul_ne_top (ENNReal.natCast_ne_top _) ENNReal.ofReal_ne_top |
| · exact hr |
| · let A : ℝ≥0∞ := (G.orbitRepresentatives hq).roots.card * |
| ENNReal.ofReal (5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1) |
| have hA : A≠∞ := ENNReal.mul_ne_top (ENNReal.natCast_ne_top _) ENNReal.ofReal_ne_top |
| have ht := ENNReal.Tendsto.const_mul (a := A) (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one |
| (show ENNReal.ofReal (1/2:ℝ)<1 by norm_num)) (Or.inr hA) |
| simpa only [mul_zero,zero_mul] using ENNReal.Tendsto.mul_const ht (Or.inr (by norm_num : (2:ℝ≥0∞)≠∞)) |
| · intro N n |
| exact G.expected_heavy_normalizedDiscrepancy_bound hc hl hq D K hD hK p (q n) N x hx |
|
|
| end BondGraph |
| end CriticalPercolation |
| end |
| end |
|
|
| end OAI |
|
|
