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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import OAI.Probability.FreeIsing.Posterior.Fields
import OAI.Probability.FreeIsing.Trees.FinitePathGibbs
namespace OAI
noncomputable section
open scoped BigOperators
namespace Problem367.PathPosterior
def interaction (beta : ℝ) (n : ℕ) (s : Fin (n + 1) → IsingSpin) : ℝ :=
beta * ∑ i : Fin n, spinSign (s i.castSucc) * spinSign (s i.succ)
def spin (n : ℕ) (s : Fin (n + 1) → IsingSpin) (v : Fin (n + 1)) : ℝ :=
spinSign (s v)
def childFields {n : ℕ} (y : Fin (n + 1) → ℝ) (i : ℕ) : ℝ :=
if hi : i < n then y ⟨i + 1, Nat.succ_lt_succ hi⟩ else 0
@[simp] theorem childFields_fin {n : ℕ} (y : Fin (n + 1) → ℝ) (i : Fin n) :
childFields y i = y i.succ := by
simp only [childFields, dite_eq_left i.isLt]
rfl
theorem energy_eq_fieldEnergy (beta : ℝ) (n : ℕ)
(y : Fin (n + 1) → ℝ) (s : Fin (n + 1) → IsingSpin) :
FinitePathGibbs.energy beta (y 0) (childFields y) n s =
FinitePosterior.fieldEnergy (interaction beta n) (spin n) y s := by
unfold FinitePathGibbs.energy FinitePosterior.fieldEnergy interaction spin
rw [Fin.sum_univ_succ]
simp only [childFields_fin, add_mul, Finset.sum_add_distrib, Finset.mul_sum, mul_assoc]
ring
theorem mean_eq_fieldMean (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ)
(f : (Fin (n + 1) → IsingSpin) → ℝ) :
FinitePathGibbs.mean beta (y 0) (childFields y) n f =
FinitePosterior.fieldMean (interaction beta n) (spin n) f y := by
unfold FinitePathGibbs.mean FinitePosterior.fieldMean
simp_rw [FinitePathGibbs.mass_eq_exp_energy]
rw [FinitePathGibbs.partition_eq_sum_exp]
simp_rw [energy_eq_fieldEnergy]
simp_rw [div_mul_eq_mul_div]
rw [← Finset.sum_div]
apply congrArg (fun a => a / ∑ s, Real.exp
(FinitePosterior.fieldEnergy (interaction beta n) (spin n) y s))
apply Finset.sum_congr rfl
intro s _
ring
def rootMean (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) : ℝ :=
FinitePosterior.fieldMean (interaction beta n) (spin n) (fun s => spinSign (s 0)) y
def rootField (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) : ℝ :=
Real.artanh (rootMean beta n y)
theorem rootMean_eq_tanh (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
rootMean beta n y =
Real.tanh (FinitePathGibbs.rootField beta (y 0) (childFields y) n) := by
rw [rootMean, ← mean_eq_fieldMean, FinitePathGibbs.mean_first_eq_tanh]
theorem rootMean_mem_Ioo (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
rootMean beta n y ∈ Set.Ioo (-1 : ℝ) 1 := by
rw [rootMean_eq_tanh]
exact ⟨Real.neg_one_lt_tanh _, Real.tanh_lt_one _⟩
theorem rootField_eq (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
rootField beta n y = FinitePathGibbs.rootField beta (y 0) (childFields y) n := by
rw [rootField, rootMean_eq_tanh, Real.artanh_tanh]
theorem endpoint_covariance (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
FinitePosterior.fieldMean (interaction beta n) (spin n)
(fun s => spinSign (s 0) * spinSign (s (Fin.last n))) y -
rootMean beta n y * FinitePosterior.fieldMean (interaction beta n) (spin n)
(fun s => spinSign (s (Fin.last n))) y =
(1 - rootMean beta n y ^ 2) * Real.tanh beta ^ n *
(∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i)) := by
simp only [rootMean, ← mean_eq_fieldMean]
exact FinitePathGibbs.endpointCovariance_eq beta (y 0) (childFields y) n
theorem hasDerivAt_rootMean_endpoint (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
HasDerivAt
(fun t => rootMean beta n (fun v => y v + if v = Fin.last n then t else 0))
((1 - rootMean beta n y ^ 2) * Real.tanh beta ^ n *
(∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i))) 0 := by
let : Nonempty (Fin (n + 1) → IsingSpin) := ⟨fun _ => .pos⟩
have hd := FinitePosterior.hasDerivAt_fieldMean_coordinate
(interaction beta n) (spin n) (fun s => spinSign (s 0)) y (Fin.last n)
change HasDerivAt _
(FinitePosterior.fieldMean (interaction beta n) (spin n)
(fun s => spinSign (s 0) * spinSign (s (Fin.last n))) y -
rootMean beta n y * FinitePosterior.fieldMean (interaction beta n) (spin n)
(fun s => spinSign (s (Fin.last n))) y) 0 at hd
rw [endpoint_covariance] at hd
exact hd
theorem hasDerivAt_rootField_endpoint (beta : ℝ) (n : ℕ) (y : Fin (n + 1) → ℝ) :
HasDerivAt
(fun t => rootField beta n (fun v => y v + if v = Fin.last n then t else 0))
(Real.tanh beta ^ n *
(∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i))) 0 := by
have hm := rootMean_mem_Ioo beta n y
have hv : 1 - rootMean beta n y ^ 2 ≠ 0 := by
have hs : rootMean beta n y ^ 2 < 1 := by
rw [rootMean_eq_tanh]
exact Real.tanh_sq_lt_one _
linarith
have hbase : (fun v => y v + if v = Fin.last n then (0 : ℝ) else 0) = y := by
funext v
simp
have ha : HasDerivAt Real.artanh (1 / (1 - rootMean beta n y ^ 2))
(rootMean beta n (fun v => y v + if v = Fin.last n then (0 : ℝ) else 0)) := by
simpa only [hbase] using hasDerivAt_artanh_interval hm
have hd := ha.comp 0 (hasDerivAt_rootMean_endpoint beta n y)
simpa only [hbase, Function.comp_def, rootField, one_div, mul_assoc,
inv_mul_cancel_left₀ hv] using hd
theorem hasDerivAt_rootField_endpoint_at (beta : ℝ) (n : ℕ)
(y : Fin (n + 1) → ℝ) (t : ℝ) :
HasDerivAt
(fun s => rootField beta n (fun v => y v + if v = Fin.last n then s else 0))
(Real.tanh beta ^ n *
(∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta)
(FinitePathGibbs.backwardField beta
(childFields (fun v => y v + if v = Fin.last n then t else 0)) n i))) t := by
have hd := hasDerivAt_rootField_endpoint beta n
(fun v => y v + if v = Fin.last n then t else 0)
have hc := hd.comp_of_eq t ((hasDerivAt_id t).sub_const t) (by simp)
simp only [mul_one, Function.comp_def, id_eq] at hc
convert! hc using 1
funext s
congr 1
funext v
split_ifs <;> ring
theorem response_mem_Icc {beta : ℝ} (hbeta : 0 ≤ beta) (n : ℕ)
(y : Fin (n + 1) → ℝ) :
(Real.tanh beta ^ n *
(∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i))) ∈
Set.Icc 0 (Real.tanh beta ^ n) := by
have hθ := tanh_nonneg_of_beta_nonneg hbeta
have hs := (Real.tanh_sq_lt_one beta).le
have ha0 : 0 ≤ ∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i) :=
Finset.prod_nonneg fun i _ => (responseAttenuation_pos hs _).le
have ha1 : (∏ i ∈ Finset.range n,
responseAttenuation (Real.tanh beta) (FinitePathGibbs.backwardField beta (childFields y) n i)) ≤ 1 :=
Finset.prod_le_one₀ (fun i _ => (responseAttenuation_pos hs _).le)
(fun i _ => responseAttenuation_le_one hs _)
exact ⟨mul_nonneg (pow_nonneg hθ n) ha0,
mul_le_of_le_one_right (pow_nonneg hθ n) ha1⟩
theorem lipschitzWith_rootField_endpoint {beta : ℝ} (hbeta : 0 ≤ beta)
(n : ℕ) (y : Fin (n + 1) → ℝ) :
LipschitzWith ⟨Real.tanh beta ^ n, pow_nonneg (tanh_nonneg_of_beta_nonneg hbeta) n⟩
(fun t => rootField beta n (fun v => y v + if v = Fin.last n then t else 0)) := by
apply lipschitzWith_of_nnnorm_deriv_le
· intro t
exact (hasDerivAt_rootField_endpoint_at beta n y t).differentiableAt
· intro t
rw [(hasDerivAt_rootField_endpoint_at beta n y t).deriv]
have hb := response_mem_Icc hbeta n
(fun v => y v + if v = Fin.last n then t else 0)
change |Real.tanh beta ^ n * _| ≤ Real.tanh beta ^ n
rw [abs_of_nonneg hb.1]
exact hb.2
end Problem367.PathPosterior
end
end OAI
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