| import Mathlib |
| import OAI.Probability.CriticalPercolation.Modular.PeakReturn |
| import OAI.Probability.CriticalPercolation.Modular.PeakRatioTransport |
| import OAI.Probability.CriticalPercolation.Modular.MarkedTiltedTransport |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology BigOperators |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| omit [Countable V] [Countable E] in |
| lemma clusterPeakRatio_aut (φ : G.Aut) (ω : Set E) (x : V) : |
| G.clusterPeakRatio (φ.edge '' ω) (φ.vertex x) = G.clusterPeakRatio ω x := by |
| unfold clusterPeakRatio |
| rw [←φ.vertex.tsum_eq] |
| simp only [G.peakRatioTransport_aut] |
|
|
| def heavyFlag (p : unitInterval) (u : E → ℝ) (x : V) : Prop := |
| G.modularMass x (G.cluster (atParameter p u) x) = ∞ |
|
|
| lemma measurable_heavyFlag (p : unitInterval) (x : V) : |
| Measurable (fun u => G.heavyFlag p u x) := |
| ((G.measurable_modularClusterMass x x).comp (measurable_atParameter p)).eq_const ∞ |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyFlag_iff_of_reachable (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| {p q : unitInterval} (hqp : q≤p) (u : E → ℝ) {x y : V} |
| (hxy : (G.openGraph (atParameter q u)).Reachable x y) : |
| G.heavyFlag p u x ↔ G.heavyFlag p u y := by |
| have hr := hxy.mono (G.openGraph_mono (atParameter_mono u hqp)) |
| unfold heavyFlag |
| rw [G.cluster_eq_of_reachable hr,G.modularMass_eq_top_change_base hc hl x y] |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyFlag_aut (φ : G.Aut) (p : unitInterval) (u : E → ℝ) (x : V) : |
| G.heavyFlag p (reindexMarks φ.edge u) (φ.vertex x) ↔ G.heavyFlag p u x := by |
| unfold heavyFlag |
| rw [atParameter_reindex,←φ.cluster_image,φ.modularMass_image] |
|
|
| def heavyRestrictedPeakTransport (p q : unitInterval) (u : E → ℝ) (x y : V) : ℝ≥0∞ := |
| if G.heavyFlag p u x then G.peakRatioTransport (atParameter q u) x y else 0 |
|
|
| lemma measurable_heavyRestrictedPeakTransport (p q : unitInterval) (x y : V) : |
| Measurable (fun u => G.heavyRestrictedPeakTransport p q u x y) := |
| ((G.measurable_peakRatioTransport x y).comp (measurable_atParameter q)).ite |
| (G.measurable_heavyFlag p x).setOf measurable_const |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyRestrictedPeakTransport_aut (p q : unitInterval) (φ : G.Aut) |
| (u : E → ℝ) (x y : V) : |
| G.heavyRestrictedPeakTransport p q (reindexMarks φ.edge u) (φ.vertex x) (φ.vertex y) = |
| G.heavyRestrictedPeakTransport p q u x y := by |
| simp only [heavyRestrictedPeakTransport,G.heavyFlag_aut,atParameter_reindex,G.peakRatioTransport_aut] |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyRestrictedPeakTransport_out (p q : unitInterval) (u : E → ℝ) (x : V) : |
| (∑' y, G.heavyRestrictedPeakTransport p q u x y) = |
| if G.heavyFlag p u x then G.clusterPeakRatio (atParameter q u) x else 0 := by |
| by_cases h : G.heavyFlag p u x <;> simp only [heavyRestrictedPeakTransport,h,ite_true,ite_false, |
| clusterPeakRatio,tsum_zero] |
| |
| omit [Countable V] [Countable E] in |
| lemma heavyRestrictedPeakTransport_swap (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| {p q : unitInterval} (hqp : q≤p) (u : E → ℝ) (x y : V) : |
| G.heavyRestrictedPeakTransport p q u x y = |
| if G.heavyFlag p u y then G.peakRatioTransport (atParameter q u) x y else 0 := by |
| by_cases hy : y∈G.clusterPeaks (atParameter q u) x |
| · simp only [heavyRestrictedPeakTransport,G.heavyFlag_iff_of_reachable hc hl hqp u hy.1] |
| · simp only [heavyRestrictedPeakTransport,peakRatioTransport,ite_eq_right hy,ite_self] |
|
|
| lemma heavyRestrictedPeakTransport_incoming_ae |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| {p q : unitInterval} (hqp : q≤p) (y : V) : |
| ∀ᵐ u ∂markLaw, (∑' x, G.heavyRestrictedPeakTransport p q u x y * G.stabilizerWeight y x) = |
| if G.heavyFlag p u y then G.peakReturn (atParameter q u) y else 0 := by |
| have ha := G.marks_ae_of_law_ae q _ (G.peakRatioTransport_incoming_ae hc hl hq q y) |
| filter_upwards [ha] with u hu |
| simp only [G.heavyRestrictedPeakTransport_swap hc hl hqp] |
| by_cases h : G.heavyFlag p u y |
| · simpa only [h,ite_true] using hu |
| · simp only [h,ite_false,zero_mul,tsum_zero] |
|
|
| lemma expected_heavy_clusterPeakRatio |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| {p q : unitInterval} (hqp : q≤p) : |
| (∑ x ∈ (G.orbitRepresentatives hq).roots, |
| ∫⁻ u, if G.heavyFlag p u x then G.clusterPeakRatio (atParameter q u) x else 0 ∂markLaw) = |
| ∑ x ∈ (G.orbitRepresentatives hq).roots, |
| ∫⁻ u, if G.heavyFlag p u x then G.peakReturn (atParameter q u) x else 0 ∂markLaw := by |
| simp_rw [←G.heavyRestrictedPeakTransport_out] |
| rw [OrbitRepresentatives.mark_tilted_transport G (G.orbitRepresentatives hq) hc hl |
| (G.heavyRestrictedPeakTransport p q) (G.measurable_heavyRestrictedPeakTransport p q) |
| (G.heavyRestrictedPeakTransport_aut p q)] |
| apply Finset.sum_congr rfl |
| intro x _ |
| exact lintegral_congr_ae (G.heavyRestrictedPeakTransport_incoming_ae hc hl hq hqp x) |
|
|
| lemma tendsto_expected_heavy_peakReturn |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q) |
| (hqp : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) (x : V) : |
| Tendsto (fun n => ∫⁻ u, if G.heavyFlag p u x then G.peakReturn (atParameter (q n) u) x else 0 ∂markLaw) |
| atTop (𝓝 0) := by |
| have hmeas (n : ℕ) : Measurable (fun u => |
| if G.heavyFlag p u x then G.peakReturn (atParameter (q n) u) x else 0) := |
| ((G.measurable_peakReturn x).comp (measurable_atParameter (q n))).ite |
| (G.measurable_heavyFlag p x).setOf measurable_const |
| have hbound (n : ℕ) : ∀ᵐ u ∂markLaw, |
| (if G.heavyFlag p u x then G.peakReturn (atParameter (q n) u) x else 0) ≤ (1:ℝ≥0∞) := by |
| exact Eventually.of_forall (fun u => by split_ifs; exact G.peakReturn_le_one _ x; exact zero_le) |
| have hae := G.marks_ae_of_law_ae p _ (G.no_heavy_finite_peaks_ae hc hl hq p) |
| have ht : ∀ᵐ u ∂markLaw, Tendsto (fun n => |
| if G.heavyFlag p u x then G.peakReturn (atParameter (q n) u) x else 0) atTop (𝓝 (0:ℝ≥0∞)) := by |
| filter_upwards [hae] with u hu |
| by_cases h : G.heavyFlag p u x |
| · simp only [h,ite_true] |
| apply G.tendsto_peakReturn_of_exhaustion _ _ |
| ((atParameter_mono u).comp hmono) (atParameter_iUnion p q hqp hlim u) x |
| by_cases hp : G.clusterPeak (atParameter p u) x |
| · right |
| intro hf |
| exact hu x hf.toFinset.card ⟨h,⟨hf.toFinset,le_rfl,fun _ hy => hf.mem_toFinset.mpr hy⟩,hp⟩ |
| · exact Or.inl hp |
| · simp only [h,ite_false] |
| exact tendsto_const_nhds |
| simpa only [lintegral_zero] using |
| tendsto_lintegral_of_dominated_convergence (μ := (markLaw : Measure (E → ℝ))) |
| (fun _ => (1:ℝ≥0∞)) hmeas hbound (by simp) ht |
|
|
| theorem tendsto_sum_expected_heavy_clusterPeakRatio |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q) |
| (hqp : ∀ n, q n ≤ p) (hlim : Tendsto q atTop (𝓝 p)) : |
| Tendsto (fun n => ∑ x ∈ (G.orbitRepresentatives hq).roots, |
| ∫⁻ u, if G.heavyFlag p u x then G.clusterPeakRatio (atParameter (q n) u) x else 0 ∂markLaw) |
| atTop (𝓝 0) := by |
| simp_rw [G.expected_heavy_clusterPeakRatio hc hl hq (hqp _)] |
| have ht := tendsto_finsetSum (G.orbitRepresentatives hq).roots |
| (fun x _ => G.tendsto_expected_heavy_peakReturn hc hl hq p q hmono hqp hlim x) |
| simpa only [Finset.sum_const_zero] using ht |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
