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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.IncidenceDegree
import OAI.Probability.CriticalPercolation.Exploration.IncidenceIdentity
import OAI.Probability.CriticalPercolation.Exploration.QueryBand
import OAI.Probability.CriticalPercolation.Exploration.QueryNormalized
import OAI.Probability.CriticalPercolation.Modular.CappedDiscrepancy
namespace OAI
section
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation
open FiniteProduct
namespace BondGraph
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
omit [Fintype V] in
lemma clusterDiscrepancy_eq_centered (p : ℝ) (hp : p≠0) (ω : Set E) (v : V) :
G.clusterDiscrepancy p ω v = -(IncidenceSearch.tree G v).centeredSum p ω / p := by
rw [IncidenceSearch.tree_centered,clusterDiscrepancy,← Finset.sum_neg_distrib,Finset.sum_div]
apply Finset.sum_congr rfl
intro e _
change (if e∈G.incidentEdges (G.cluster ω v) then
(if e∈ω then -(1-p)/p else 1) else 0) =
-(if e∈G.incidentEdges (G.cluster ω v) then (if e∈ω then 1-p else -p) else 0)/p
by_cases he : e∈G.incidentEdges (G.cluster ω v)
· by_cases hω : e∈ω <;> simp [he,hω,hp]
· simp [he]
lemma root_discrepancy_bound (p : ℝ) (hp : p ∈ Set.Ioo 0 1)
(w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b=1) (hn : ∀ e b, 0≤w e b)
(hf : LazyDFS.Fresh w p ∅) (D : ℕ) (hDpos : 1≤D)
(hD : ∀ v, (G.incidentEdges {v}).toFinset.card ≤ D) (K : ℕ) (v : V) :
expect w (fun ω => G.rootMass (cappedMass (4^K)) ω v * |G.clusterDiscrepancy p ω v|) ≤
(D:ℝ)/p*(6/(2:ℝ)^K) := by
let t := IncidenceSearch.tree G v
have hwel : ∀ ω : Set E, G.rootMass (cappedMass (4^K)) ω v * |G.clusterDiscrepancy p ω v| ≤
(D:ℝ)/p * (|t.centeredSum p ω| * min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ))) := by
intro ω
rw [G.clusterDiscrepancy_eq_centered p hp.1.ne' ω v,abs_div,abs_neg,abs_of_pos hp.1]
change G.rootMass (cappedMass (4^K)) ω v * (|t.centeredSum p ω|/p) ≤ _
by_cases hT : t.queryCount ω = 0
· rw [t.centeredSum_eq_zero_of_count ω hT p]
simp
· have hTD : t.queryCount ω ≤ D*G.clusterSize ω v := by
rw [IncidenceSearch.tree_count]
simpa only [clusterSize,mul_comm] using G.incident_card_le (G.cluster ω v) D hD
have hh := G.capped_root_le_normalized (4^K) (t.queryCount ω) D
(by positivity) (Nat.pos_of_ne_zero hT) hDpos ω v hTD
have hh' : G.rootMass (cappedMass (4^K)) ω v ≤ (D:ℝ)*
min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ)) := by
simpa only [Nat.cast_pow,Nat.cast_ofNat] using hh
calc
_ ≤ ((D:ℝ)*min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ))) *
(|t.centeredSum p ω|/p) := mul_le_mul_of_nonneg_right hh'
(div_nonneg (abs_nonneg _) hp.1.le)
_ = _ := by ring
calc
_ ≤ expect w (fun ω => (D:ℝ)/p *
(|t.centeredSum p ω| * min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ)))) := by
exact expect_mono hn (fun ω => hwel ω)
_ = (D:ℝ)/p * expect w (fun ω =>
|t.centeredSum p ω| * min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ))) := expect_const_mul _ _ _
_ ≤ _ := mul_le_mul_of_nonneg_left
(t.normalized_centered_bound ∅ (IncidenceSearch.tree_valid G v) p
⟨hp.1.le,hp.2.le⟩ w hw hn hf K) (div_nonneg (Nat.cast_nonneg D) hp.1.le)
theorem finite_averaged_adjacent (p : ℝ) (hp : p ∈ Set.Ioo 0 1)
(w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b=1) (hn : ∀ e b, 0≤w e b)
(h0 : ∀ e, w e False=1-p) (h1 : ∀ e, w e True=p)
(D : ℕ) (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).toFinset.card ≤ D)
(x y : E → V) (he : ∀ e, G.ends e = s(x e,y e)) (K : ℕ) :
(∑ e, prob w (BinaryLaw.endpointEvent G (x e) (y e) (4^K))) ≤
(Fintype.card V : ℝ)*((D:ℝ)/p*(6/(2:ℝ)^K)) := by
have hf : LazyDFS.Fresh w p ∅ := fun e _ => ⟨h0 e,h1 e⟩
calc
_ ≤ ∑ e, expect w (fun ω => G.edgeDiscrepancy (cappedMass (4^K)) p ω e) := by
apply Finset.sum_le_sum
intro e _
exact G.endpoint_prob_le_discrepancy p hp w hw hn e (x e) (y e) (he e)
(h0 e) (h1 e) (4^K) (by positivity)
_ = expect w (fun ω => ∑ v, G.rootMass (cappedMass (4^K)) ω v * G.clusterDiscrepancy p ω v) := by
erw [← expect_sum]
congr 1
funext ω
exact G.sum_edgeDiscrepancy _ p ω
_ ≤ expect w (fun ω => ∑ v, G.rootMass (cappedMass (4^K)) ω v * |G.clusterDiscrepancy p ω v|) := by
apply expect_mono hn
intro ω
apply Finset.sum_le_sum
intro v _
apply mul_le_mul_of_nonneg_left (le_abs_self _)
exact div_nonneg (cappedMass_nonneg _ _) (Nat.cast_nonneg _)
_ = ∑ v, expect w (fun ω => G.rootMass (cappedMass (4^K)) ω v * |G.clusterDiscrepancy p ω v|) := by
erw [expect_sum]
_ ≤ ∑ _v : V, (D:ℝ)/p*(6/(2:ℝ)^K) := by
exact Finset.sum_le_sum (fun v _ => G.root_discrepancy_bound p hp w hw hn hf D hDpos hD K v)
_ = _ := by simp
end BondGraph
end CriticalPercolation
end
end
end
end OAI
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