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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.WeightedPeakTransport
import OAI.Probability.CriticalPercolation.Modular.GeometricPeakBand
import OAI.Probability.CriticalPercolation.Modular.NormalizedDiscrepancy
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal BigOperators Topology
namespace CriticalPercolation
lemma ennreal_geometric_band (v : ℝ≥0∞) (hv : v≠∞) (N : ℕ)
(hN : ENNReal.ofReal ((16:ℝ)^N)≤v) :
∃ m : ℕ, ENNReal.ofReal ((16:ℝ)^(N+m))≤v ∧ v<ENNReal.ofReal ((16:ℝ)^(N+m+1)) := by
have hNr : (16:ℝ)^N≤v.toReal := (ENNReal.ofReal_le_iff_le_toReal hv).mp hN
have hone : 1≤v.toReal := (one_le_pow₀ (by norm_num : (1:ℝ)≤16)).trans hNr
obtain ⟨k,hk,hk'⟩ := exists_nat_pow_near hone (by norm_num : (1:ℝ)<16)
have hNk : N≤k := by
by_contra h
have hkle : k+1≤N := by omega
have hp : (16:ℝ)^(k+1)≤16^N := pow_le_pow_right₀ (by norm_num) hkle
exact (not_lt_of_ge hNr) (hk'.trans_le hp)
obtain ⟨m,rfl⟩ := Nat.exists_eq_add_of_le hNk
exact ⟨m,ENNReal.ofReal_le_of_le_toReal hk,(ENNReal.lt_ofReal_iff_toReal_lt hv).mpr hk'⟩
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E) [Countable V] [Countable E]
omit [Countable V] [Countable E] in
lemma normalizedClusterVolume_ne_top (ω : Set E) (x : V) : G.normalizedClusterVolume ω x≠∞ := by
unfold normalizedClusterVolume
split_ifs with h
· exact ENNReal.zero_ne_top
· exact ENNReal.inv_ne_top.mpr h
def discrepancyVolumeTail (p : ℝ) (T : ℝ≥0∞) (ω : Set E) (x : V) : ℝ≥0∞ :=
if T≤G.normalizedClusterVolume ω x then G.normalizedAbsoluteDiscrepancy p ω x else 0
lemma measurable_discrepancyVolumeTail (p : ℝ) (T : ℝ≥0∞) (x : V) :
Measurable (fun ω : Set E => G.discrepancyVolumeTail p T ω x) :=
(G.measurable_normalizedAbsoluteDiscrepancy p x).ite
(measurableSet_le measurable_const (G.measurable_normalizedClusterVolume x)) measurable_const
omit [Countable V] [Countable E] in
lemma discrepancyVolumeTail_le_bands (p : ℝ) (N : ℕ) (ω : Set E) (x : V) :
G.discrepancyVolumeTail p (ENNReal.ofReal ((16:ℝ)^N)) ω x ≤
∑' m : ℕ, G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+m)))
(ENNReal.ofReal ((16:ℝ)^(N+m+1))) ω x := by
unfold discrepancyVolumeTail
split_ifs with h
· obtain ⟨m,hm⟩ := ennreal_geometric_band _ (G.normalizedClusterVolume_ne_top ω x) N h
calc
_ = G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+m)))
(ENNReal.ofReal ((16:ℝ)^(N+m+1))) ω x := (ite_eq_left hm).symm
_ ≤ _ := ENNReal.le_tsum (f := fun index : ℕ =>
G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+index)))
(ENNReal.ofReal ((16:ℝ)^(N+index+1))) ω x) m
· exact zero_le
lemma expected_discrepancyTail_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 : unitInterval) (N : ℕ)
(x : V) (hx : x∈(G.orbitRepresentatives hq).roots) :
(∫⁻ ω, G.discrepancyVolumeTail p (ENNReal.ofReal ((16:ℝ)^N)) ω x ∂G.law p) ≤
(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 A : ℝ := 5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1
let R : ℝ≥0∞ := (G.orbitRepresentatives hq).roots.card
calc
_ ≤ ∫⁻ ω, ∑' m : ℕ, G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+m)))
(ENNReal.ofReal ((16:ℝ)^(N+m+1))) ω x ∂G.law p :=
lintegral_mono (fun ω => G.discrepancyVolumeTail_le_bands p N ω x)
_ = ∑' m : ℕ, ∫⁻ ω, G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+m)))
(ENNReal.ofReal ((16:ℝ)^(N+m+1))) ω x ∂G.law p :=
lintegral_tsum (fun m => (G.measurable_discrepancyVolumeBand p _ _ x).aemeasurable)
_ ≤ ∑' m : ℕ, R * ENNReal.ofReal (A*(1/2:ℝ)^(N+m)) := by
apply ENNReal.tsum_le_tsum
intro m
exact (Finset.single_le_sum (f := fun y => ∫⁻ ω,
G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^(N+m)))
(ENNReal.ofReal ((16:ℝ)^(N+m+1))) ω y ∂G.law p) (fun _ _ => zero_le) hx).trans
(G.expected_discrepancyBand_geometric hc hl hq D K hD hK p (N+m))
_ = _ := by
have he (m : ℕ) : ENNReal.ofReal (A*(1/2:ℝ)^(N+m)) =
ENNReal.ofReal A * (ENNReal.ofReal (1/2:ℝ))^N * (ENNReal.ofReal (1/2:ℝ))^m := by
rw [ENNReal.ofReal_mul (show 0≤A by dsimp [A]; positivity),
ENNReal.ofReal_pow (by norm_num : (0:ℝ)≤1/2),pow_add,mul_assoc]
simp_rw [he,←mul_assoc]
rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric]
norm_num [R,A,ENNReal.ofReal_div_of_pos]
end BondGraph
end CriticalPercolation
end
end
end OAI
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