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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ClusterTransportGeometry
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical BigOperators Topology
namespace CriticalPercolation.FiniteClusters
variable {V E : Type*} (G : BondGraph V E)
def weight (r s : ℕ) (ω : Set E) (v : V) : ℝ :=
if small G r ω v then mass G s ω v else 0
def transport (r s : ℕ) (p : ℝ) (ω : Set E) (v : V) (e : E) : ℝ :=
if e ∈ G.incidentEdges (G.cluster ω v) then weight G r s ω v * coefficient p ω e else 0
def clusterWeight (r s : ℕ) (ω : Set E) (v : V) : ℝ :=
if small G r ω v then cappedMass s (size G ω v) else 0
def truncatedEdgeMass (r s : ℕ) (ω : Set E) (e : E) : ℝ :=
clusterWeight G r s ω (G.ends e).out.1 +
if (G.openGraph ω).Reachable (G.ends e).out.1 (G.ends e).out.2 then 0
else clusterWeight G r s ω (G.ends e).out.2
lemma weight_reachable (r s : ℕ) {ω : Set E} {u v : V}
(h : (G.openGraph ω).Reachable u v) : weight G r s ω u = weight G r s ω v := by
rw [weight,weight,small_reachable G r h,mass_eq_of_reachable G s h]
lemma weight_nonneg (r s : ℕ) (ω : Set E) (v : V) : 0 ≤ weight G r s ω v := by
unfold weight
split_ifs
· exact mass_nonneg G s ω v
· rfl
lemma clusterWeight_nonneg (r s : ℕ) (ω : Set E) (v : V) : 0 ≤ clusterWeight G r s ω v := by
unfold clusterWeight
split_ifs
· exact cappedMass_nonneg _ _
· rfl
lemma clusterWeight_le (r s : ℕ) (ω : Set E) (v : V) :
clusterWeight G r s ω v ≤ cappedMass s (size G ω v) := by
unfold clusterWeight
split_ifs
· rfl
· exact cappedMass_nonneg _ _
lemma weighted_cluster_summable (hl : G.LocallyFinite) (r s : ℕ) (ω : Set E) (x : V) :
Summable (fun u => if u ∈ G.cluster ω x then weight G r s ω u else 0) := by
by_cases hs : small G r ω x
· apply summable_of_ne_finset_zero (s := (small_finite G hl hs).toFinset)
intro u hu
simpa only [Set.Finite.mem_toFinset,not_false_eq_true] using
(ite_eq_right (show u ∉ G.cluster ω x from fun h => hu ((small_finite G hl hs).mem_toFinset.mpr h)) :
(if u ∈ G.cluster ω x then weight G r s ω u else 0) = 0)
· have hz : (fun u => if u ∈ G.cluster ω x then weight G r s ω u else 0) = 0 := by
funext u
by_cases hu : u ∈ G.cluster ω x
· rw [ite_eq_left hu,← weight_reachable G r s hu,weight,ite_eq_right hs]
rfl
· simp [hu]
rw [hz]
exact summable_zero
lemma tsum_weighted_cluster (hl : G.LocallyFinite) (r s : ℕ) (ω : Set E) (x : V) :
(∑' u, if u ∈ G.cluster ω x then weight G r s ω u else 0) = clusterWeight G r s ω x := by
by_cases hs : small G r ω x
· have hC := small_finite G hl hs
have hpos : 0 < size G ω x := (Set.ncard_pos hC).mpr ⟨x,SimpleGraph.Reachable.refl x⟩
rw [tsum_eq_sum (s := hC.toFinset) (fun u hu => ite_eq_right (fun h => hu (hC.mem_toFinset.mpr h)))]
calc
_ = ∑ _u ∈ hC.toFinset, mass G s ω x := by
apply Finset.sum_congr rfl
intro u hu
rw [ite_eq_left (hC.mem_toFinset.mp hu),← weight_reachable G r s (hC.mem_toFinset.mp hu),weight,ite_eq_left hs]
_ = _ := by
rw [Finset.sum_const,nsmul_eq_mul,← Set.ncard_eq_toFinset_card _ hC]
change (size G ω x : ℝ) * (cappedMass s (size G ω x) / size G ω x) = _
rw [mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hpos.ne')]
exact (ite_eq_left hs).symm
· rw [clusterWeight,ite_eq_right hs]
calc
_ = ∑' _u : V, (0:ℝ) := by
apply tsum_congr
intro u
by_cases hu : u ∈ G.cluster ω x
· rw [ite_eq_left hu,← weight_reachable G r s hu,weight,ite_eq_right hs]
· rw [ite_eq_right hu]
_ = 0 := tsum_zero
lemma transport_sum_out (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) :
(∑' e, transport G r s p ω u e) = if small G r ω u then rootDiscrepancy G s p ω u else 0 := by
have hf : (fun e => transport G r s p ω u e) =
(fun e => weight G r s ω u * (if e ∈ G.incidentEdges (G.cluster ω u) then coefficient p ω e else 0)) := by
funext e
unfold transport
split_ifs <;> simp
rw [hf,tsum_mul_left]
unfold weight rootDiscrepancy rootSum
split_ifs <;> simp
lemma transport_sum_in (hl : G.LocallyFinite) (r s : ℕ) (p : ℝ) (ω : Set E) (e : E) :
(∑' u, transport G r s p ω u e) = coefficient p ω e * truncatedEdgeMass G r s ω e := by
let x := (G.ends e).out.1
let y := (G.ends e).out.2
have hi (u : V) : e ∈ G.incidentEdges (G.cluster ω u) ↔
u ∈ G.cluster ω x ∪ G.cluster ω y := by
exact Set.ext_iff.mp (incidentRoots G ω e) u
by_cases hxy : (G.openGraph ω).Reachable x y
· have heq := G.cluster_eq_of_reachable hxy
have hf : (fun u => transport G r s p ω u e) =
fun u => coefficient p ω e * (if u ∈ G.cluster ω x then weight G r s ω u else 0) := by
funext u
unfold transport
rw [hi,← heq,Set.union_self]
split_ifs <;> ring
rw [hf,tsum_mul_left,tsum_weighted_cluster G hl]
simp only [truncatedEdgeMass,show (G.openGraph ω).Reachable (G.ends e).out.1 (G.ends e).out.2 from hxy,ite_true,add_zero]
rfl
· have hd := G.cluster_disjoint_of_not_reachable hxy
have hf : (fun u => transport G r s p ω u e) = fun u => coefficient p ω e *
((if u ∈ G.cluster ω x then weight G r s ω u else 0) +
(if u ∈ G.cluster ω y then weight G r s ω u else 0)) := by
funext u
unfold transport
rw [hi]
by_cases hx : u ∈ G.cluster ω x
· have hy : u ∉ G.cluster ω y := fun hy => Set.disjoint_left.mp hd hx hy
simp only [hx,hy,Set.mem_union,true_or,ite_true,ite_false,add_zero]
ring
· by_cases hy : u ∈ G.cluster ω y
· simp only [hx,hy,Set.mem_union,or_true,ite_true,ite_false,zero_add]; ring
· simp only [hx,hy,Set.mem_union,or_false,ite_false,add_zero,mul_zero]
rw [hf,tsum_mul_left,(weighted_cluster_summable G hl r s ω x).tsum_add
(weighted_cluster_summable G hl r s ω y),tsum_weighted_cluster G hl,tsum_weighted_cluster G hl]
simp only [truncatedEdgeMass,show ¬(G.openGraph ω).Reachable (G.ends e).out.1 (G.ends e).out.2 from hxy,ite_false]
rfl
lemma truncatedEdgeMass_nonneg (r s : ℕ) (ω : Set E) (e : E) :
0 ≤ truncatedEdgeMass G r s ω e := by
unfold truncatedEdgeMass
split_ifs
· simpa using clusterWeight_nonneg G r s ω (G.ends e).out.1
· exact add_nonneg (clusterWeight_nonneg G r s ω _) (clusterWeight_nonneg G r s ω _)
lemma truncatedEdgeMass_le (r s : ℕ) (ω : Set E) (e : E) :
truncatedEdgeMass G r s ω e ≤ edgeMass G s ω e := by
unfold truncatedEdgeMass edgeMass
split_ifs
· simpa using clusterWeight_le G r s ω (G.ends e).out.1
· exact add_le_add (clusterWeight_le G r s ω _) (clusterWeight_le G r s ω _)
lemma transport_sum_in_bound (hl : G.LocallyFinite) (r s : ℕ) {p : ℝ}
(hp : p ∈ Ioo 0 1) (ω : Set E) (e : E) :
|∑' u, transport G r s p ω u e| ≤ 2/p := by
rw [transport_sum_in G hl,abs_mul,abs_of_nonneg (truncatedEdgeMass_nonneg G r s ω e)]
calc
_ ≤ (1/p)*2 := mul_le_mul (coefficient_abs_le hp ω e)
((truncatedEdgeMass_le G r s ω e).trans (edgeMass_le_two G s ω e))
(truncatedEdgeMass_nonneg G r s ω e) (div_nonneg zero_le_one hp.1.le)
_ = _ := by ring
end CriticalPercolation.FiniteClusters
end
end
end OAI
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