File size: 7,179 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ClusterTransportSums
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)
lemma transport_row_zero (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) (e : E)
(he : e ∉ G.incidentEdges (G.vertexBall u r)) : transport G r s p ω u e = 0 := by
unfold transport weight
by_cases hs : small G r ω u
· have hi : e ∉ G.incidentEdges (G.cluster ω u) := by
rintro ⟨v,hv,hve⟩
exact he ⟨v,small_cluster_subset G hs hv,hve⟩
rw [ite_eq_right hi]
· simp only [hs,ite_false,zero_mul,ite_self]
lemma transport_column_zero (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) (e : E)
(hu : u ∉ G.vertexBall (G.ends e).out.1 r ∪ G.vertexBall (G.ends e).out.2 r) :
transport G r s p ω u e = 0 := by
unfold transport weight
by_cases hs : small G r ω u
· have hi : e ∉ G.incidentEdges (G.cluster ω u) := by
intro he
have he' : u ∈ G.cluster ω (G.ends e).out.1 ∪ G.cluster ω (G.ends e).out.2 :=
Set.ext_iff.mp (incidentRoots G ω e) u |>.mp he
rcases he' with h | h
· exact hu (Or.inl (G.vertexBall_comm.mp (small_cluster_subset G hs h.symm)))
· exact hu (Or.inr (G.vertexBall_comm.mp (small_cluster_subset G hs h.symm)))
rw [ite_eq_right hi]
· simp only [hs,ite_false,zero_mul,ite_self]
lemma mass_le_one (s : ℕ) (ω : Set E) (v : V) : mass G s ω v ≤ 1 := by
by_cases hn : size G ω v = 0
· simp [mass,hn]
· have hp : (0:ℝ) < size G ω v := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)
apply (div_le_one hp).mpr
exact (cappedMass_le_one _ _).trans (by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hn)
lemma weight_le_one (r s : ℕ) (ω : Set E) (v : V) : weight G r s ω v ≤ 1 := by
unfold weight
split_ifs
· exact mass_le_one G s ω v
· norm_num
lemma transport_bound (r s : ℕ) {p : ℝ} (hp : p ∈ Ioo 0 1)
(ω : Set E) (u : V) (e : E) : |transport G r s p ω u e| ≤ 1/p := by
unfold transport
split_ifs
· rw [abs_mul,abs_of_nonneg (weight_nonneg G r s ω u)]
exact (mul_le_mul (weight_le_one G r s ω u) (coefficient_abs_le hp ω e)
(abs_nonneg _) (by norm_num)).trans_eq (one_mul _)
· simpa only [abs_zero] using div_nonneg (show (0:ℝ)≤1 by norm_num) hp.1.le
lemma transport_aut (φ : G.Aut) (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) (e : E) :
transport G r s p (φ.edge '' ω) (φ.vertex u) (φ.edge e) = transport G r s p ω u e := by
simp only [transport,weight,incidentCluster_aut G φ,small_aut G φ,mass_aut G φ,coefficient_aut G φ]
variable [Countable V] [Countable E]
lemma measurable_weight (r s : ℕ) (u : V) : Measurable (fun ω => weight G r s ω u) :=
(measurable_mass G s u).ite (measurable_small G r u).setOf measurable_const
lemma measurable_transport (r s : ℕ) (p : ℝ) (u : V) (e : E) :
Measurable (fun ω => transport G r s p ω u e) :=
((measurable_weight G r s u).mul (measurable_coefficient p e)).ite
(measurable_incidentCluster G u e) measurable_const
lemma integrable_transport (r s : ℕ) (p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1) (u : V) (e : E) :
Integrable (fun ω => transport G r s p ω u e) (G.law p) := by
apply Integrable.mono' (integrable_const (1/(p:ℝ))) (measurable_transport G r s p u e).aestronglyMeasurable
exact Eventually.of_forall (fun ω => transport_bound G r s hp ω u e)
def expectedTransport (r s : ℕ) (p : unitInterval) (u : V) (e : E) : ℝ :=
∫ ω, transport G r s p ω u e ∂G.law p
lemma expectedTransport_aut (φ : G.Aut) (r s : ℕ) (p : unitInterval) (u : V) (e : E) :
expectedTransport G r s p (φ.vertex u) (φ.edge e) = expectedTransport G r s p u e := by
unfold expectedTransport
conv_lhs => rw [← G.law_map_aut p φ]
rw [integral_map (BernoulliProduct.measurable_image_equiv φ.edge).aemeasurable
(measurable_transport G r s p (φ.vertex u) (φ.edge e)).aestronglyMeasurable]
apply integral_congr_ae
exact Eventually.of_forall (fun ω => transport_aut G φ r s p ω u e)
omit [Countable V] [Countable E] in
lemma expectedTransport_row_zero (r s : ℕ) (p : unitInterval) (u : V) (e : E)
(he : e ∉ G.incidentEdges (G.vertexBall u r)) : expectedTransport G r s p u e = 0 := by
unfold expectedTransport
simp only [transport_row_zero G r s p _ u e he,integral_zero]
omit [Countable V] [Countable E] in
lemma expectedTransport_column_zero (r s : ℕ) (p : unitInterval) (u : V) (e : E)
(hu : u ∉ G.vertexBall (G.ends e).out.1 r ∪ G.vertexBall (G.ends e).out.2 r) :
expectedTransport G r s p u e = 0 := by
unfold expectedTransport
simp only [transport_column_zero G r s p _ u e hu,integral_zero]
omit [Countable V] [Countable E] in
lemma expectedTransport_row_finite (hl : G.LocallyFinite) (r s : ℕ) (p : unitInterval) (u : V) :
(Function.support (expectedTransport G r s p u)).Finite := by
apply (G.finite_incident hl (G.vertexBall_finite hl u r)).subset
intro e he
by_contra hn
exact he (expectedTransport_row_zero G r s p u e hn)
omit [Countable V] [Countable E] in
lemma expectedTransport_column_finite (hl : G.LocallyFinite) (r s : ℕ) (p : unitInterval) (e : E) :
(Function.support (fun u => expectedTransport G r s p u e)).Finite := by
apply ((G.vertexBall_finite hl (G.ends e).out.1 r).union
(G.vertexBall_finite hl (G.ends e).out.2 r)).subset
intro u hu
by_contra hn
exact hu (expectedTransport_column_zero G r s p u e hn)
lemma expectedTransport_sum_out (hl : G.LocallyFinite) (r s : ℕ) (p : unitInterval)
(hp : (p:ℝ) ∈ Ioo 0 1) (u : V) :
(∑' e, expectedTransport G r s p u e) = ∫ ω, ∑' e, transport G r s p ω u e ∂G.law p := by
have hI := G.finite_incident hl (G.vertexBall_finite hl u r)
rw [tsum_eq_sum (s := hI.toFinset) (fun e he => expectedTransport_row_zero G r s p u e
(fun h => he (hI.mem_toFinset.mpr h)))]
unfold expectedTransport
rw [← integral_finsetSum _ (fun e _ => integrable_transport G r s p hp u e)]
apply integral_congr_ae
apply Eventually.of_forall
intro ω
exact (tsum_eq_sum (s := hI.toFinset) (fun e he => transport_row_zero G r s p ω u e
(fun h => he (hI.mem_toFinset.mpr h)))).symm
lemma expectedTransport_sum_in (hl : G.LocallyFinite) (r s : ℕ) (p : unitInterval)
(hp : (p:ℝ) ∈ Ioo 0 1) (e : E) :
(∑' u, expectedTransport G r s p u e) = ∫ ω, ∑' u, transport G r s p ω u e ∂G.law p := by
have hB := (G.vertexBall_finite hl (G.ends e).out.1 r).union
(G.vertexBall_finite hl (G.ends e).out.2 r)
rw [tsum_eq_sum (s := hB.toFinset) (fun u hu => expectedTransport_column_zero G r s p u e
(fun h => hu (hB.mem_toFinset.mpr h)))]
unfold expectedTransport
rw [← integral_finsetSum _ (fun u _ => integrable_transport G r s p hp u e)]
apply integral_congr_ae
apply Eventually.of_forall
intro ω
exact (tsum_eq_sum (s := hB.toFinset) (fun u hu => transport_column_zero G r s p ω u e
(fun h => hu (hB.mem_toFinset.mpr h)))).symm
end CriticalPercolation.FiniteClusters
end
end
end OAI
|