File size: 2,004 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphActionCoordinates
import OAI.Probability.CriticalPercolation.Modular.ActionMassTransport
import OAI.Probability.CriticalPercolation.Modular.Unimodularity
namespace OAI
section
noncomputable section
open Set
open scoped Classical BigOperators
namespace CriticalPercolation.BondGraph
variable {V E Λ : Type*} [Group Λ] [MulAction Λ V] [MulAction Λ E]
(G : BondGraph V E)
def unimodularRoot_of_action (ha : G.PreservesEnds (Λ := Λ))
(S : Set V) (hS : S.Finite) (hcover : ∀ x, ∃ u∈S, x∈MulAction.orbit Λ u)
(o : V) (w : V → ℝ) (hp : ∀ x, 0<w x)
(hw : ∀ (φ : Λ) x, w (φ • x) = w x)
(hfin : ∀ x y, (ActionOrbitTransport.stabilizerOrbit Λ V x y).Finite)
(hbal : ∀ x y, w x * (ActionOrbitTransport.suborbitCard Λ V x y : ℝ) =
w y * (ActionOrbitTransport.suborbitCard Λ V y x : ℝ)) : G.UnimodularRoot := by
let R := ActionOrbitTransport.representatives Λ V S hS hcover
let Z := ∑ x∈R.roots, w x
have hZ : 0<Z := by
obtain ⟨u,hu,_,_⟩ := R.cover o
exact Finset.sum_pos' (fun x _ => (hp x).le) ⟨u,hu,hp u⟩
let W := fun x => w x / Z
have hW : ∀ (φ : Λ) x, W (φ • x) = W x := by
intro φ x
simp only [W,hw]
have hB (x y : V) : W x * (ActionOrbitTransport.suborbitCard Λ V x y : ℝ) =
W y * (ActionOrbitTransport.suborbitCard Λ V y x : ℝ) := by
simp only [W,div_mul_eq_mul_div,hbal]
refine ⟨R.roots,W,?_,?_,?_,?_⟩
· intro x _
exact div_pos (hp x) hZ
· change (∑ x∈R.roots,w x/Z)=1
rw [←Finset.sum_div,div_self (ne_of_gt hZ)]
· intro v
obtain ⟨u,hu,φ,hφ⟩ := R.cover v
exact ⟨u,hu,G.smulAut ha φ,hφ⟩
· intro F hF hr hc
exact ActionOrbitTransport.Representatives.massTransport Λ V R hfin W hW hB F
(fun φ x y => hF (G.smulAut ha φ) x y) hr hc
end CriticalPercolation.BondGraph
end
end
end OAI
|