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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 | 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
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