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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 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 | import OAI.Probability.ClassicalON.BlockOrbit
namespace OAI
universe uE uV
noncomputable section
open MeasureTheory
open scoped InnerProductSpace BigOperators Classical
namespace ClassicalON
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E]
def blockCoupling (k : ℕ) (left right : E → V) (b : E → ℝ) (s : V → Spin (3+k)) : E → ℝ :=
fun e => b e*blockRadius k (s (left e))*blockRadius k (s (right e))
def blockTailEnergy (k : ℕ) (left right : E → V) (b : E → ℝ) (s : V → Spin (3+k)) : ℝ :=
∑ e,b e*⟪blockTail k (s (left e)),blockTail k (s (right e))⟫_ℝ
omit [Fintype V] [Fintype E] in
theorem blockCoupling_bounds (k : ℕ) (left right : E → V) (b : E → ℝ) (hb : ∀ e,0 ≤ b e)
(s : V → Spin (3+k)) (e : E) : 0 ≤ blockCoupling k left right b s e ∧ blockCoupling k left right b s e ≤ b e := by
unfold blockCoupling
constructor
· exact mul_nonneg (mul_nonneg (hb e) (blockRadius_nonneg _ _)) (blockRadius_nonneg _ _)
· calc
_ ≤ b e*1*1 := mul_le_mul (mul_le_mul_of_nonneg_left (blockRadius_le_one _ _) (hb e))
(blockRadius_le_one _ _) (blockRadius_nonneg _ _) (by simpa using hb e)
_ = b e := by ring
omit [Fintype V] in
theorem blockFiberConfiguration_energy (k : ℕ) (left right : E → V) (b : E → ℝ)
(s : V → Spin (3+k)) (t : V → Spin 3) :
freeSpinEnergy (3+k) left right b (blockFiberConfiguration k s t)=
freeSpinEnergy 3 left right (blockCoupling k left right b s) t+blockTailEnergy k left right b s := by
unfold freeSpinEnergy blockTailEnergy blockFiberConfiguration
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro e _
rw [blockFiber_inner]
unfold blockCoupling
ring
omit [Fintype V] [Fintype E] in
theorem blockFiberConfiguration_first (k : ℕ) (x y : V) (s : V → Spin (3+k)) (t : V → Spin 3) :
coordinateSpinProduct (3+k) ⟨0,by omega⟩ x y (blockFiberConfiguration k s t)=
(blockRadius k (s x)*blockRadius k (s y))*firstSpinProduct x y t := by
unfold coordinateSpinProduct blockFiberConfiguration
rw [← blockHead_first,← blockHead_first,blockHead_fiber,blockHead_fiber]
simp only [PiLp.smul_apply,smul_eq_mul,firstSpinProduct]
ring
def freeSpinPartition (n : ℕ) (left right : E → V) (b : E → ℝ) : ℝ :=
∫ s,Real.exp (freeSpinEnergy n left right b s) ∂freeSpinReference n
theorem freeSpinPartition_pos (n : ℕ) [NeZero n] (left right : E → V) (b : E → ℝ) :
0 < freeSpinPartition n left right b :=
compact_integral_pos (continuous_freeSpinEnergy _ _ _ _).rexp (fun _ => Real.exp_pos _)
theorem block_fiber_partition (k : ℕ) (left right : E → V) (b : E → ℝ) (s : V → Spin (3+k)) :
(∫ t,Real.exp (freeSpinEnergy (3+k) left right b (blockFiberConfiguration k s t)) ∂freeSpinReference 3)=
Real.exp (blockTailEnergy k left right b s)*freeSpinPartition 3 left right (blockCoupling k left right b s) := by
simp_rw [blockFiberConfiguration_energy,Real.exp_add]
rw [integral_mul_const]
exact mul_comm _ _
theorem block_fiber_numerator (k : ℕ) (left right : E → V) (b : E → ℝ)
(x y : V) (s : V → Spin (3+k)) :
(∫ t,Real.exp (freeSpinEnergy (3+k) left right b (blockFiberConfiguration k s t))*
coordinateSpinProduct (3+k) ⟨0,by omega⟩ x y (blockFiberConfiguration k s t) ∂freeSpinReference 3)=
(Real.exp (blockTailEnergy k left right b s)*freeSpinPartition 3 left right (blockCoupling k left right b s))*
((blockRadius k (s x)*blockRadius k (s y))*
freeSpinMean 3 left right (blockCoupling k left right b s) (firstSpinProduct x y)) := by
have hz := (freeSpinPartition_pos 3 left right (blockCoupling k left right b s)).ne'
simp_rw [blockFiberConfiguration_energy,Real.exp_add,blockFiberConfiguration_first]
have he (t : V → Spin 3) :
Real.exp (freeSpinEnergy 3 left right (blockCoupling k left right b s) t)*
Real.exp (blockTailEnergy k left right b s)*
(blockRadius k (s x)*blockRadius k (s y)*firstSpinProduct x y t)=
(Real.exp (blockTailEnergy k left right b s)*(blockRadius k (s x)*blockRadius k (s y)))*
(Real.exp (freeSpinEnergy 3 left right (blockCoupling k left right b s) t)*firstSpinProduct x y t) := by ring
simp_rw [he]
rw [integral_const_mul]
unfold freeSpinMean weightedMean
change _=(_*freeSpinPartition 3 left right (blockCoupling k left right b s))*
(_*(_/freeSpinPartition 3 left right (blockCoupling k left right b s)))
field_simp
end ClassicalON
end
end OAI
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