| 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 |
|
|
