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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Probability.FreeIsing.Model
namespace OAI
noncomputable section
namespace Problem367
@[simp] theorem isingSpin_neg_ne_pos : IsingSpin.neg ≠ IsingSpin.pos := by decide
@[simp] theorem isingSpin_pos_ne_neg : IsingSpin.pos ≠ IsingSpin.neg := by decide
private theorem spinKernel_univ : (Finset.univ : Finset IsingSpin) = {.neg, .pos} := by decide
theorem sum_isingSpin {α : Type*} [AddCommMonoid α] (f : IsingSpin → α) :
∑ a : IsingSpin, f a = f .neg + f .pos := by
simp [spinKernel_univ]
def spinSign : IsingSpin → ℝ
| .neg => -1
| .pos => 1
@[simp] theorem spinSign_neg : spinSign .neg = -1 := rfl
@[simp] theorem spinSign_pos : spinSign .pos = 1 := rfl
@[simp] theorem spinSign_sq (a : IsingSpin) : spinSign a ^ 2 = 1 := by
cases a <;> norm_num [spinSign]
@[simp] theorem spinSign_mul_self (a : IsingSpin) : spinSign a * spinSign a = 1 := by
simpa only [pow_two] using spinSign_sq a
theorem spinSign_mul (a b : IsingSpin) :
spinSign a * spinSign b = if a = b then 1 else -1 := by
cases a <;> cases b <;> norm_num [spinSign]
def transitionKernel (theta : ℝ) (a b : IsingSpin) : ℝ :=
(1 + theta * spinSign a * spinSign b) / 2
theorem transitionKernel_eq (theta : ℝ) (a b : IsingSpin) :
transitionKernel theta a b = (1 + (if a = b then theta else -theta)) / 2 := by
cases a <;> cases b <;> simp [transitionKernel, spinSign]
theorem transitionKernel_symm (theta : ℝ) (a b : IsingSpin) :
transitionKernel theta a b = transitionKernel theta b a := by
unfold transitionKernel
ring
@[simp] theorem transitionKernel_zero (a b : IsingSpin) :
transitionKernel 0 a b = (1 / 2 : ℝ) := by
simp [transitionKernel]
@[simp] theorem transitionKernel_one (a b : IsingSpin) :
transitionKernel 1 a b = if a = b then 1 else 0 := by
cases a <;> cases b <;> norm_num [transitionKernel, spinSign]
@[simp] theorem sum_spinSign : ∑ a : IsingSpin, spinSign a = 0 := by
simp [sum_isingSpin, spinSign]
@[simp] theorem transitionKernel_sum (theta : ℝ) (a : IsingSpin) :
∑ b : IsingSpin, transitionKernel theta a b = 1 := by
rw [sum_isingSpin]
simp only [transitionKernel, spinSign_neg, spinSign_pos]
ring
@[simp] theorem transitionKernel_spinSum (theta : ℝ) (a : IsingSpin) :
∑ b : IsingSpin, transitionKernel theta a b * spinSign b = theta * spinSign a := by
rw [sum_isingSpin]
simp only [transitionKernel, spinSign_neg, spinSign_pos]
ring
theorem transitionKernel_comp (s t : ℝ) (a c : IsingSpin) :
∑ b : IsingSpin, transitionKernel s a b * transitionKernel t b c =
transitionKernel (s * t) a c := by
rw [sum_isingSpin]
simp only [transitionKernel, spinSign_neg, spinSign_pos]
ring
theorem transitionKernel_nonneg {theta : ℝ} (hlo : -1 ≤ theta) (hhi : theta ≤ 1)
(a b : IsingSpin) : 0 ≤ transitionKernel theta a b := by
rw [transitionKernel_eq]
split_ifs <;> linarith
theorem transitionKernel_le_one {theta : ℝ} (hlo : -1 ≤ theta) (hhi : theta ≤ 1)
(a b : IsingSpin) : transitionKernel theta a b ≤ 1 := by
rw [transitionKernel_eq]
split_ifs <;> linarith
def transitionKernelIter (theta : ℝ) : ℕ → IsingSpin → IsingSpin → ℝ
| 0, a, b => if a = b then 1 else 0
| n + 1, a, c => ∑ b : IsingSpin,
transitionKernelIter theta n a b * transitionKernel theta b c
theorem transitionKernelIter_eq (theta : ℝ) (n : ℕ) (a b : IsingSpin) :
transitionKernelIter theta n a b = transitionKernel (theta ^ n) a b := by
induction n generalizing a b with
| zero => simp [transitionKernelIter]
| succ n ih =>
simp only [transitionKernelIter, ih, transitionKernel_comp, pow_succ]
@[simp] theorem transitionKernelIter_sum (theta : ℝ) (n : ℕ) (a : IsingSpin) :
∑ b : IsingSpin, transitionKernelIter theta n a b = 1 := by
simp only [transitionKernelIter_eq, transitionKernel_sum]
theorem transitionKernelIter_spinSum (theta : ℝ) (n : ℕ) (a : IsingSpin) :
∑ b : IsingSpin, transitionKernelIter theta n a b * spinSign b =
theta ^ n * spinSign a := by
simp only [transitionKernelIter_eq, transitionKernel_spinSum]
theorem transitionKernel_stationary (theta : ℝ) (b : IsingSpin) :
∑ a : IsingSpin, (1 / 2 : ℝ) * transitionKernel theta a b = 1 / 2 := by
rw [sum_isingSpin]
simp only [transitionKernel, spinSign_neg, spinSign_pos]
ring
theorem transitionKernelIter_correlation (theta : ℝ) (n : ℕ) :
∑ a : IsingSpin, ∑ b : IsingSpin,
(1 / 2 : ℝ) * transitionKernelIter theta n a b * spinSign a * spinSign b =
theta ^ n := by
simp only [transitionKernelIter_eq, sum_isingSpin, transitionKernel,
spinSign_neg, spinSign_pos]
ring
theorem freeCylinderWeight_eq_transitionKernel {d : ℕ} (theta : ℝ)
(D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) :
freeCylinderWeight theta D eta = ENNReal.ofReal
((1 / 2 : ℝ) * ∏ e ∈ inducedEdges D, transitionKernel theta (eta e.1) (eta e.2)) := by
simp only [freeCylinderWeight, transitionKernel_eq]
end Problem367
end
end OAI
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