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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.QueryBand
import OAI.Probability.CriticalPercolation.Exploration.IncidenceIdentity
import OAI.Probability.CriticalPercolation.Exploration.QueryNormalized
import OAI.Probability.CriticalPercolation.Modular.ClusterMassInfinite
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical BigOperators
namespace CriticalPercolation.FiniteClusters
open FiniteProduct
variable {V E : Type*} (G : BondGraph V E)
lemma size_pos {ω : Set E} {v : V} (hC : (G.cluster ω v).Finite) :
0 < size G ω v := (Set.ncard_pos hC).mpr ⟨v,SimpleGraph.Reachable.refl v⟩
lemma mass_eq_min (s : ℕ) (hs : 0<s) (ω : Set E) (v : V)
(hC : (G.cluster ω v).Finite) :
mass G s ω v = min (1/(s:ℝ)) (1/(size G ω v : ℝ)) := by
have hm : (0:ℝ)<size G ω v := Nat.cast_pos.mpr (size_pos G hC)
have hs' : (0:ℝ)<s := Nat.cast_pos.mpr hs
unfold mass cappedMass
by_cases hms : (size G ω v : ℝ)/s ≤ 1
· rw [min_eq_left hms,min_eq_left]
· field_simp
· apply (div_le_div_iff₀ hs' hm).mpr
have hh := (div_le_one hs').mp hms
simpa only [one_mul] using hh
· have hh : (s:ℝ) ≤ size G ω v := (one_le_div hs').mp (le_of_lt (lt_of_not_ge hms))
rw [min_eq_right (le_of_lt (lt_of_not_ge hms)),min_eq_right]
apply (div_le_div_iff₀ hm hs').mpr
simpa only [one_mul] using hh
lemma mass_le_normalized (s T D : ℕ) (hs : 0<s) (hT : 0<T) (hD : 1≤D)
(ω : Set E) (v : V) (hC : (G.cluster ω v).Finite) (hTD : T ≤ D*size G ω v) :
mass G s ω v ≤ (D:ℝ)*min (1/(s:ℝ)) (1/(T:ℝ)) := by
rw [mass_eq_min G s hs ω v hC,mul_min_of_nonneg _ _ (Nat.cast_nonneg D)]
apply le_min
· exact (min_le_left _ _).trans
(le_mul_of_one_le_left (by positivity) (Nat.one_le_cast.mpr hD))
· apply (min_le_right _ _).trans
rw [mul_one_div]
apply (div_le_div_iff₀ (Nat.cast_pos.mpr (size_pos G hC)) (Nat.cast_pos.mpr hT)).mpr
simpa only [one_mul,Nat.cast_mul] using (Nat.cast_le.mpr hTD : (T:ℝ)≤(D*size G ω v : ℕ))
variable [Fintype E]
lemma rootSum_eq_centered (p : ℝ) (hp : p≠0) (ω : Set E) (v : V) :
rootSum G p ω v = -(IncidenceSearch.tree G v).centeredSum p ω / p := by
rw [IncidenceSearch.tree_centered,rootSum,tsum_fintype,← Finset.sum_neg_distrib,Finset.sum_div]
apply Finset.sum_congr rfl
intro e _
by_cases he : e∈G.incidentEdges (G.cluster ω v)
· by_cases hω : e∈ω <;> simp [he,hω,hp,coefficient]
· simp [he]
lemma finite_rootAbsolute_expect (p : ℝ) (hp : p ∈ 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}).ncard ≤ D) (K : ℕ) (v : V) :
expect w (fun ω => rootAbsolute G (4^K) p ω v) ≤ (D:ℝ)/p*(6/(2:ℝ)^K) := by
let t := IncidenceSearch.tree G v
have hwel (ω : Set E) : rootAbsolute G (4^K) p ω v ≤
(D:ℝ)/p * (|t.centeredSum p ω| * min (1/(4:ℝ)^K) (1/(t.queryCount ω : ℝ))) := by
rw [rootAbsolute,rootSum_eq_centered G p hp.1.ne' ω v,abs_div,abs_neg,abs_of_pos hp.1]
by_cases hT : t.queryCount ω = 0
· rw [t.centeredSum_eq_zero_of_count ω hT p]
simp
· have hC := G.finite_cluster_of_finite_bonds (Set.toFinite ω) v
have hTD : t.queryCount ω ≤ D*size G ω v := by
rw [IncidenceSearch.tree_count,← Set.ncard_eq_toFinset_card']
simpa only [size,mul_comm] using G.incident_ncard_le _ hC D (fun w _ => hD w)
have hh := mass_le_normalized G (4^K) (t.queryCount ω) D
(by positivity) (Nat.pos_of_ne_zero hT) hDpos ω v hC hTD
have hh' : mass G (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)))) := expect_mono hn hwel
_ = (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)
lemma finite_rootAbsolute_integral (p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1)
(D : ℕ) (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) (K : ℕ) (v : V) :
(∫ ω, rootAbsolute G (4^K) p ω v ∂G.law p) ≤ (D:ℝ)/p*(6/(2:ℝ)^K) := by
rw [BondGraph.law,← BernoulliProduct.law_const,BernoulliProduct.integral_law_eq_expect]
exact finite_rootAbsolute_expect G p hp _
(fun _ => BernoulliProduct.bitWeight_sum p) (fun _ => BernoulliProduct.bitWeight_nonneg p)
(fun _ _ => ⟨by simp [BernoulliProduct.bitWeight],by simp [BernoulliProduct.bitWeight]⟩)
D hDpos hD K v
end CriticalPercolation.FiniteClusters
end
end
end OAI
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