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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Clusters.ExpectedCluster
import OAI.Probability.CriticalPercolation.Clusters.ComponentSelf
import OAI.Probability.CriticalPercolation.ProductMeasure.Distribution
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace CriticalPercolation.BondGraph
open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion
open scoped Classical
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
lemma expectedResidual_alt (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s : V) (F : Set V → ℝ) (U : Set V) :
G.expectedResidual w S s F U =
expect w (fun ω => if s ∉ roots (delete (G.openGraph ω) U) S then
G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S) -
F (component (delete (G.openGraph ω) U) s) else 0) := by
have h := G.finite_spatialMarkov w hw S U
(fun W H => if s ∉ W then G.expectedCluster w s F W -
F (component (delete H U) s) else 0)
have hdel : ∀ H W, delete (delete H W) U = delete H (W ∪ U) := by
intro H W
rw [delete_delete, Set.union_comm]
simp_rw [hdel] at h
have hleft : (fun ω : E → Prop => if s ∉ roots (delete (G.openGraph ω) U) S then
G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S) -
F (component (delete (G.openGraph ω) (roots (delete (G.openGraph ω) U) S ∪ U)) s)
else 0) =
(fun ω => if s ∉ roots (delete (G.openGraph ω) U) S then
G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S) -
F (component (delete (G.openGraph ω) U) s) else 0) := by
funext ω
split_ifs with hs
· rfl
· rw [delete_roots_union_component _ S U s hs]
rw [hleft] at h
erw [h]
unfold expectedResidual
congr 1
funext ω
split_ifs
· rw [expect_zero]
· rw [expect_sub, expect_const w hw]
rfl
lemma later_cluster_residual (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s t : V) (F g : Set V → ℝ) (Q : Set V → Prop)
(hQ : ∀ U, Q U → Disjoint U S ∧ s ∉ U) :
(expect w (fun ω => if Q (component (G.openGraph ω) t) then
if s ∉ roots (G.openGraph ω) S then
g (component (G.openGraph ω) t) *
(G.expectedCluster w s F (roots (G.openGraph ω) S) -
F (component (G.openGraph ω) s)) else 0 else 0)) =
expect w (fun ω => if Q (component (G.openGraph ω) t) then
g (component (G.openGraph ω) t) *
G.expectedResidual w S s F (component (G.openGraph ω) t) else 0) := by
have h := G.finite_spatialMarkov w hw {t} ∅
(fun U H => if Q U then if s ∉ roots H S then
g U * (G.expectedCluster w s F (roots H S) - F (component H s)) else 0 else 0)
simp only [delete_empty, roots_singleton] at h
have hleft : (fun ω : E → Prop => if Q (component (G.openGraph ω) t) then
if s ∉ roots (delete (G.openGraph ω) (component (G.openGraph ω) t)) S then
g (component (G.openGraph ω) t) *
(G.expectedCluster w s F (roots (delete (G.openGraph ω) (component (G.openGraph ω) t)) S) -
F (component (delete (G.openGraph ω) (component (G.openGraph ω) t)) s))
else 0 else 0) =
(fun ω => if Q (component (G.openGraph ω) t) then
if s ∉ roots (G.openGraph ω) S then
g (component (G.openGraph ω) t) *
(G.expectedCluster w s F (roots (G.openGraph ω) S) - F (component (G.openGraph ω) s))
else 0 else 0) := by
funext ω
by_cases hq : Q (component (G.openGraph ω) t)
· simp only [ite_eq_left hq]
obtain ⟨hd, hs⟩ := hQ _ hq
rw [roots_delete_component _ S t ((not_mem_roots_iff_disjoint _ S t).mpr hd), component_delete_component _ t s hs]
· simp only [ite_eq_right hq]
rw [hleft] at h
erw [h]
congr 1
funext ω
by_cases hq : Q (component (G.openGraph ω) t)
· simp only [ite_eq_left hq]
rw [G.expectedResidual_alt w hw]
rw [← expect_const_mul]
congr 1
funext ω'
split_ifs <;> simp
· simp only [ite_eq_right hq, expect_zero]
end CriticalPercolation.BondGraph
end
end
end OAI
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