File size: 2,301 Bytes
9b6e014 | 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 | import OAI.Probability.ClassicalON.SpinReindex
namespace OAI
universe uE uF uK uV uW
noncomputable section
open MeasureTheory Set
open scoped BigOperators Classical
namespace ClassicalON
variable {V : Type uV} {W : Type uW} {E : Type uE} {F : Type uF} {K : Type uK} [Fintype V] [Fintype W] [Fintype E] [Fintype F] [Fintype K]
omit [Fintype K] in
theorem blockBondMean_local (l r : E → V) (b : E → ℝ) (B : Set V) (label : V → K)
(elabel : E → K) (k : K) (l' r' : F → W) (b' : F → ℝ) (B' : Set W)
(ev : BlockVertex B label k ≃ ↥(B'ᶜ)) (ee : {e // elabel e=k} ≃ F)
(hb : ∀ e,b' (ee e)=b e.val)
(hl : ∀ (s : ↥(B'ᶜ) → Spin 3) e,
glueFamily B' (fun _ => poleSpin true) s (l' (ee e))=
blockSpin B label k (fun v => s (ev v)) (l e.val))
(hr : ∀ (s : ↥(B'ᶜ) → Spin 3) e,
glueFamily B' (fun _ => poleSpin true) s (r' (ee e))=
blockSpin B label k (fun v => s (ev v)) (r e.val))
(f : (F → Bool) → ℝ) :
poleBondMean l' r' b' B' (fun _ => true) f=
blockBondMean l r b B label elabel k (fun η => f (arrowEquiv ee η)) := by
rw [poleBondMean_collapse]
have he (s : ↥(B'ᶜ) → Spin 3) :
freeSpinEnergy 3 l' r' b' (glueFamily B' (fun _ => poleSpin true) s)=
blockEnergy l r b B label elabel k (fun v => s (ev v)) :=
freeSpinEnergy_edgeEquiv 3 ee _ _ _ _ _ _ _ _ hb (hl s) (hr s)
have hs (s : ↥(B'ᶜ) → Spin 3) :
(∑ η,spinBondWeight l' r' b' (glueFamily B' (fun _ => poleSpin true) s) η*f η)=
∑ η,blockBondWeight l r b B label elabel k (fun v => s (ev v)) η*f (arrowEquiv ee η) := by
have h := sum_spinBondWeight_edgeEquiv ee (fun e => l e.val) (fun e => r e.val) l' r'
(fun e => b e.val) b' (blockSpin B label k (fun v => s (ev v)))
(glueFamily B' (fun _ => poleSpin true) s) hb (hl s) (hr s)
(fun η => f (arrowEquiv ee η))
simp only [arrowEquiv,Equiv.coe_fn_mk,Equiv.apply_symm_apply] at h
convert h using 1; congr!
simp_rw [he,hs]
unfold blockBondMean
convert weightedMean_pi_reindex ev (fun _ => sphereProbability 3)
(fun s => Real.exp (blockEnergy l r b B label elabel k s))
(fun s => ∑ η,blockBondWeight l r b B label elabel k s η*f (arrowEquiv ee η)) using 1
end ClassicalON
end
end OAI
|