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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteInterpolationMeasure
import OAI.Probability.CriticalPercolation.Criticality.SharpnessPhi
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 clusterTail_iff_size {ω : Set E} {v : V} (hC : (G.cluster ω v).Finite) (s : ℕ) :
ω ∈ G.clusterTail v s ↔ s ≤ size G ω v := by
constructor
· rintro ⟨A,hA,hAC,hs⟩
exact hs.trans (Set.ncard_le_ncard hAC hC)
· intro hs
exact ⟨G.cluster ω v,hC,Subset.rfl,hs⟩
lemma size_insert_disconnected (ω : Set E) (e : E) (x y : V)
(he : G.ends e = s(x,y)) (hxy : ¬(G.openGraph ω).Reachable x y)
(hx : (G.cluster ω x).Finite) (hy : (G.cluster ω y).Finite) :
size G (insert e ω) x = size G ω x + size G ω y := by
unfold size
rw [G.cluster_insert_endpoint ω e x y he,
Set.ncard_union_eq (G.cluster_disjoint_of_not_reachable hxy) hx hy]
lemma size_insert_connected (ω : Set E) (e : E) (x y : V)
(he : G.ends e = s(x,y)) (hxy : (G.openGraph ω).Reachable x y) :
size G (insert e ω) x = size G ω x := by
unfold size
rw [G.cluster_insert_endpoint ω e x y he,G.cluster_eq_of_reachable hxy,Set.union_self]
lemma edgeMass_gap (s : ℕ) (hs : 0<s) (ω : Set E) (e : E)
(hx : (G.cluster ω (G.ends e).out.1).Finite)
(hy : (G.cluster ω (G.ends e).out.2).Finite) :
(if ω ∈ G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s then (1:ℝ) else 0) ≤
edgeMass G s ω e - edgeMass G s (insert e ω) e := by
have he : G.ends e = s((G.ends e).out.1,(G.ends e).out.2) := (G.ends e).out_eq.symm
have hopen := LazyDFS.prop_openEdge_reachable G (insert e ω) (Set.mem_insert e ω) he
unfold edgeMass
rw [ite_eq_left hopen,add_zero]
by_cases hc : (G.openGraph ω).Reachable (G.ends e).out.1 (G.ends e).out.2
· rw [ite_eq_left hc,size_insert_connected G ω e _ _ he hc]
simp [BondGraph.twoClusterEvent,hc]
· rw [ite_eq_right hc,size_insert_disconnected G ω e _ _ he hc hx hy]
have hh := cappedMass_gap s (size G ω (G.ends e).out.1) (size G ω (G.ends e).out.2) hs
simpa only [BondGraph.twoClusterEvent,Set.mem_inter_iff,Set.mem_ofPred_eq,
clusterTail_iff_size G hx,clusterTail_iff_size G hy,hc,not_false_eq_true,and_true,true_and] using hh
variable [Fintype E]
lemma finite_expect_edgeDiscrepancy (s : ℕ) (p : ℝ) (hp : p ∈ Ioo 0 1)
(w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b=1)
(e : E) (h0 : w e False=1-p) (h1 : w e True=p) :
expect w (fun ω => edgeDiscrepancy G s p ω e) =
expect w (fun ω => if ¬ω e then edgeMass G s ω e - edgeMass G s (insert e ω) e else 0) := by
let : DecidableEq E := Classical.decEq E
have hflip := expect_closed_flip_decidable w hw e hp h0 h1 (fun ω => edgeMass G s ω e / p)
have hu (ω : Set E) : (Function.update ω e True : Set E) = insert e ω := by
ext a
change Function.update (fun a => ω a) e True a ↔ a=e ∨ a∈ω
by_cases ha : a=e
· subst a; simp [Function.update]
· simp only [Function.update_of_ne ha,ha,false_or]; rfl
have hleft : (fun ω : Set E => if ¬ω e then p*(edgeMass G s (Function.update ω e True) e / p) else 0) =
(fun ω => if ¬ω e then edgeMass G s (insert e ω) e else 0) := by
funext ω
rw [hu]
split_ifs <;> field_simp [hp.1.ne']
erw [hleft] at hflip
have heq (ω : Set E) : edgeDiscrepancy G s p ω e =
(if ¬ω e then edgeMass G s ω e else 0) -
(if ω e then (1-p)*(edgeMass G s ω e / p) else 0) := by
unfold edgeDiscrepancy coefficient
change (if ω e then -(1-p)/p else 1) * edgeMass G s ω e = _
by_cases hb : ω e <;> simp only [hb,ite_true,ite_false,not_true_eq_false,not_false_eq_true,sub_zero,zero_sub,one_mul]
ring
have hfunc := funext heq
erw [hfunc,expect_sub,← hflip,← expect_sub]
congr 1
funext ω
by_cases hb : ω e <;> simp [hb]
lemma finite_endpoint_le_expect (s : ℕ) (hs : 0<s) (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)
(e : E) (h0 : w e False=1-p) (h1 : w e True=p) :
prob w (fun ω => ω ∈ G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s) ≤
expect w (fun ω => edgeDiscrepancy G s p ω e) := by
rw [finite_expect_edgeDiscrepancy G s p hp w hw e h0 h1]
apply expect_mono hn
intro ω
by_cases hb : ω e
· have hconn := LazyDFS.prop_openEdge_reachable G ω hb (G.ends e).out_eq.symm
have hno : ¬ (ω : Set E) ∈ G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s := by
intro hh
exact hh.1.1 hconn
change (if (ω : Set E) ∈ G.twoClusterEvent _ _ s then (1:ℝ) else 0) ≤ _
simp only [hno,ite_false,hb,not_true_eq_false,le_refl]
· simp only [hb,not_false_eq_true,ite_true]
exact edgeMass_gap G s hs ω e (G.finite_cluster_of_finite_bonds (Set.toFinite ω) _)
(G.finite_cluster_of_finite_bonds (Set.toFinite ω) _)
lemma finite_endpoint_le_integral (s : ℕ) (hs : 0<s) (p : unitInterval)
(hp : (p:ℝ) ∈ Ioo 0 1) (e : E) :
(G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s) ≤
∫ ω, edgeDiscrepancy G s p ω e ∂G.law p := by
have hh := finite_endpoint_le_expect G s hs p hp (fun _ => BernoulliProduct.bitWeight p)
(fun _ => BernoulliProduct.bitWeight_sum p) (fun _ => BernoulliProduct.bitWeight_nonneg p)
e (by simp [BernoulliProduct.bitWeight]) (by simp [BernoulliProduct.bitWeight])
rw [BondGraph.law,← BernoulliProduct.law_const,BernoulliProduct.integral_law_eq_expect]
convert hh using 1
· exact (Sharpness.probability_eq_real p
(fun ω => ω ∈ G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s)).symm
· rfl
end CriticalPercolation.FiniteClusters
end
end
end OAI
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