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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ClusterMassRestriction
import OAI.Probability.CriticalPercolation.Modular.ClusterTransportFinite
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical BigOperators Topology
namespace CriticalPercolation.FiniteClusters
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
omit [Countable V] [Countable E] in
lemma transport_sum_out_le (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) :
(∑' e, transport G r s p ω u e) ≤ rootAbsolute G s p ω u := by
rw [transport_sum_out]
split_ifs
· exact (le_abs_self _).trans_eq (abs_rootDiscrepancy G s p ω u)
· exact rootAbsolute_nonneg G s p ω u
omit [Countable V] [Countable E] in
lemma transport_sum_out_abs (r s : ℕ) (p : ℝ) (ω : Set E) (u : V) :
|∑' e, transport G r s p ω u e| ≤ rootAbsolute G s p ω u := by
rw [transport_sum_out]
split_ifs
· exact (abs_rootDiscrepancy G s p ω u).le
· exact (abs_zero).le.trans (rootAbsolute_nonneg G s p ω u)
lemma integrable_transport_out (hl : G.LocallyFinite) (D : ℕ)
(hD : ∀ u, (G.incidentEdges {u}).ncard ≤ D) (r s : ℕ) (p : unitInterval)
(hp : (p:ℝ) ∈ Ioo 0 1) (u : V) :
Integrable (fun ω => ∑' e, transport G r s p ω u e) (G.law p) := by
apply Integrable.mono' (integrable_const ((D:ℝ)/(p:ℝ)))
(Measurable.tsum (fun e => measurable_transport G r s p u e)).aestronglyMeasurable
exact Eventually.of_forall (fun ω => (transport_sum_out_abs G r s p ω u).trans
(rootAbsolute_bound G hl D hD s hp ω u))
lemma expectedTransport_out_bound (hl : G.LocallyFinite) (D : ℕ) (hDpos : 1≤D)
(hD : ∀ u, (G.incidentEdges {u}).ncard ≤ D) (r k : ℕ) (p : unitInterval)
(hp : (p:ℝ) ∈ Ioo 0 1) (hpc : p < G.criticalProbability) (u : V) :
(∑' e, expectedTransport G r (4^k) p u e) ≤ (D:ℝ)/p*(6/(2:ℝ)^k) := by
rw [expectedTransport_sum_out G hl r (4^k) p hp u]
refine (integral_mono_ae (integrable_transport_out G hl D hD r (4^k) p hp u) ?_
(Eventually.of_forall (fun ω => transport_sum_out_le G r (4^k) p ω u))).trans
(subcritical_rootAbsolute_bound G hl D hDpos hD p hp hpc k u)
apply Integrable.mono' (integrable_const ((D:ℝ)/(p:ℝ)))
(measurable_rootAbsolute G (4^k) p u).aestronglyMeasurable
exact Eventually.of_forall (fun ω => by
rw [Real.norm_eq_abs,abs_of_nonneg (rootAbsolute_nonneg G (4^k) p ω u)]
exact rootAbsolute_bound G hl D hD (4^k) hp ω u)
omit [Countable V] [Countable E] in
lemma eventually_truncatedEdgeMass (hc : G.fullGraph.Connected) {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) (s : ℕ) :
∀ᶠ r in atTop, truncatedEdgeMass G r s ω e = edgeMass G s ω e := by
filter_upwards [eventually_small G hc hx,eventually_small G hc hy] with r hr hs
simp only [truncatedEdgeMass,clusterWeight,hr,hs,ite_true,edgeMass]
lemma expectedTransport_in_tendsto (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(s : ℕ) (p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1) (hpc : p < G.criticalProbability) (e : E) :
Tendsto (fun r => ∑' u, expectedTransport G r s p u e) atTop
(𝓝 (∫ ω, edgeDiscrepancy G s p ω e ∂G.law p)) := by
simp only [expectedTransport_sum_in G hl _ s p hp e]
apply tendsto_integral_of_dominated_convergence (fun _ => 2/(p:ℝ))
· intro r
exact (Measurable.tsum (fun u => measurable_transport G r s p u e)).aestronglyMeasurable
· exact integrable_const _
· intro r
exact Eventually.of_forall (fun ω => transport_sum_in_bound G hl r s hp ω e)
· filter_upwards [ae_finite_clusters G p hpc] with ω hω
apply tendsto_const_nhds.congr'
filter_upwards [eventually_truncatedEdgeMass G hc (hω _) (hω _) s] with r hr
rw [transport_sum_in G hl,hr]
rfl
end CriticalPercolation.FiniteClusters
end
end
end OAI
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