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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Criticality.GeometricFourth
import OAI.Probability.CriticalPercolation.Modular.NormalizedDiscrepancy
import OAI.Probability.CriticalPercolation.Modular.PeakVolumeBand
import OAI.Probability.CriticalPercolation.Modular.WeightedPeakBand
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E) [Countable V] [Countable E]
lemma expected_peakBand_le (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hq : G.QuasiTransitive) (p : unitInterval) (T U : ℝ) (hT : 0<T)
(x : V) (hx : x∈(G.orbitRepresentatives hq).roots) :
(∫⁻ ω, G.peakVolumeBand (ENNReal.ofReal T) (ENNReal.ofReal U) ω x ∂G.law p) ≤
ENNReal.ofReal ((G.orbitRepresentatives hq).roots.card/T) := by
rw [ENNReal.ofReal_div_of_pos hT,ENNReal.ofReal_natCast]
apply (ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.ofReal_pos.mpr hT).ne') (Or.inl ENNReal.ofReal_ne_top)).mpr
rw [mul_comm]
apply le_trans _ (G.expected_peakVolumeBand_mul_le hc hl hq p (ENNReal.ofReal T) (ENNReal.ofReal U))
apply mul_le_mul_right
exact Finset.single_le_sum (f := fun y => ∫⁻ ω,
G.peakVolumeBand (ENNReal.ofReal T) (ENNReal.ofReal U) ω y ∂G.law p)
(fun _ _ => zero_le) hx
lemma expected_discrepancy_peakBand_geometric (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, ENNReal.ofReal |G.weightedDiscrepancy p ω x| *
G.peakVolumeBand (ENNReal.ofReal ((16:ℝ)^n)) (ENNReal.ofReal ((16:ℝ)^(n+1))) ω x ∂G.law p) ≤
ENNReal.ofReal ((5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1)*
(1/2:ℝ)^n) := by
let C : ℝ := (D*(2*K)^2 : ℕ)
let N : ℝ := (G.orbitRepresentatives hq).roots.card
have ht : (0:ℝ)<16^n := by positivity
have h4 := G.expected_discrepancy_peakBand_fourth hc hl D K hD hK p
((16:ℝ)^n) ((16:ℝ)^(n+1)) ht (by positivity) x
apply geometric_fourth_bound _ (5*(C*16)^2*N^3) (by dsimp [N]; positivity) n
apply h4.trans
calc
_ ≤ ENNReal.ofReal (5*(C*(16:ℝ)^(n+1))^2)*
(ENNReal.ofReal (N/(16:ℝ)^n))^3 := by
apply mul_le_mul_right
exact ENNReal.pow_le_pow_left (G.expected_peakBand_le hc hl hq p _ _ ht x hx)
_ = ENNReal.ofReal ((5*(C*16)^2*N^3)/(16:ℝ)^n) := by
rw [←ENNReal.ofReal_pow (by dsimp [N]; positivity),←ENNReal.ofReal_mul (by positivity)]
congr 1
rw [pow_succ]
field_simp
ring
lemma expected_discrepancyBand_geometric (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∈(G.orbitRepresentatives hq).roots, ∫⁻ ω,
G.discrepancyVolumeBand p (ENNReal.ofReal ((16:ℝ)^n))
(ENNReal.ofReal ((16:ℝ)^(n+1))) 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)*
(1/2:ℝ)^n) := by
rw [G.expected_discrepancyBand_peak hc hl hq p _ _ (ENNReal.ofReal_pos.mpr (by positivity))]
calc _ ≤ ∑ x∈(G.orbitRepresentatives hq).roots,
ENNReal.ofReal ((5*((D*(2*K)^2 : ℕ)*16)^2*((G.orbitRepresentatives hq).roots.card : ℝ)^3+1)*
(1/2:ℝ)^n) := by
exact Finset.sum_le_sum (fun x hx => G.expected_discrepancy_peakBand_geometric hc hl hq D K hD hK p n x hx)
_ = _ := by simp [nsmul_eq_mul]
end CriticalPercolation.BondGraph
end
end
end OAI
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