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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.TiltedTransportENNReal
import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneBernoulli
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
omit [Countable V] [Countable E] in
lemma expected_mark_transport_aut
(F : (E → ℝ) → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun u => F u x y))
(ha : ∀ (φ : G.Aut) u x y,
F (reindexMarks φ.edge u) (φ.vertex x) (φ.vertex y) = F u x y)
(φ : G.Aut) (x y : V) :
(∫⁻ u, F u (φ.vertex x) (φ.vertex y) ∂markLaw) = ∫⁻ u, F u x y ∂markLaw := by
conv_lhs => rw [←markLaw_map_reindex φ.edge]
rw [lintegral_map (hm _ _) (measurable_reindexMarks φ.edge)]
simp only [ha]
omit [Countable E] in
lemma OrbitRepresentatives.mark_tilted_transport (R : G.OrbitRepresentatives)
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(F : (E → ℝ) → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun u => F u x y))
(ha : ∀ (φ : G.Aut) u x y,
F (reindexMarks φ.edge u) (φ.vertex x) (φ.vertex y) = F u x y) :
(∑ x ∈ R.roots, ∫⁻ u, ∑' y, F u x y ∂markLaw) =
∑ y ∈ R.roots, ∫⁻ u, ∑' x, F u x y * G.stabilizerWeight y x ∂markLaw := by
simp_rw [lintegral_tsum (fun y => (hm _ y).aemeasurable),
lintegral_tsum (fun x => ((hm x _).mul_const _).aemeasurable),
lintegral_mul_const _ (hm _ _)]
exact OrbitRepresentatives.tilted_massTransport_ennreal G R hc hl _
(G.expected_mark_transport_aut F hm ha)
omit [Countable V] [Countable E] in
lemma marks_ae_of_law_ae (p : unitInterval) (P : Set E → Prop) (h : ∀ᵐ ω ∂G.law p, P ω) :
∀ᵐ u ∂markLaw, P (atParameter p u) := by
apply ae_of_ae_map (measurable_atParameter p).aemeasurable
rwa [markLaw_map_atParameter,BernoulliProduct.law_const]
end CriticalPercolation.BondGraph
end
end
end OAI
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