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f205175 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | 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
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