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import OAI.Probability.FreeIsing.Gaussian.Attenuation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace Problem367.GaussianAttenuation
def pathTail (i : ℕ) (x : ℕ → ℝ) : ℕ → ℝ := fun j => x (i + 1 + j)
def pathProduct (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
(x : ℕ → ℝ) : ℝ :=
∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x)) ^ 2
@[fun_prop] lemma measurable_pathTail (i : ℕ) : Measurable (pathTail i) := by
exact Measurable.of_eval (fun index => measurable_pi_apply _)
lemma measurable_pathProduct (θ : ℝ) (n : ℕ)
{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
Measurable (pathProduct θ n shift) := by
unfold pathProduct
apply Finset.measurable_fun_prod
intro i hi
have h := (measurable_pi_apply i).add ((hs i).comp (measurable_pathTail i))
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
lemma pathProduct_nonneg (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
(x : ℕ → ℝ) : 0 ≤ pathProduct θ n shift x := by
exact Finset.prod_nonneg (fun i _ => sq_nonneg _)
lemma pathProduct_le_one {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
(shift : ℕ → (ℕ → ℝ) → ℝ) (x : ℕ → ℝ) : pathProduct θ n shift x ≤ 1 := by
exact Finset.prod_le_one₀ (fun index _ => sq_nonneg _)
(fun i _ => attenuation_sq_le_one hθ _)
lemma integrable_pathProduct_comp {Ω : Type*} [MeasurableSpace Ω]
{P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ}
(hX : ∀ i, Measurable (X i)) {θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
Integrable (fun ω => pathProduct θ n shift (fun i => X i ω)) P := by
apply (integrable_const (1 : ℝ)).mono'
((measurable_pathProduct θ n hs).comp (Measurable.of_eval hX)).aestronglyMeasurable
exact Filter.Eventually.of_forall fun ω => by
dsimp only [Function.comp_def]
rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
exact pathProduct_le_one hθ _ _ _
lemma pathProduct_succ (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
(x : ℕ → ℝ) :
pathProduct θ (n + 1) shift x =
responseAttenuation θ (x 0 + shift 0 (pathTail 0 x)) ^ 2 *
pathProduct θ n (fun i => shift (i + 1)) (pathTail 0 x) := by
unfold pathProduct
rw [Finset.prod_range_succ', mul_comm]
congr 1
apply Finset.prod_congr rfl
intro i hi
have hh : pathTail 0 x i = x (i + 1) := by
simp only [pathTail, Nat.zero_add, Nat.add_comm]
have ht : pathTail i (pathTail 0 x) = pathTail (i + 1) x := by
funext j
simp only [pathTail]
congr 1
omega
rw [hh, ht]
lemma head_indep_tail {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{X : ℕ → Ω → ℝ} (hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P) :
IndepFun (X 0) (fun ω j => X (j + 1) ω) P := by
have h := indepFun_of_disjoint_blocks hind hX ({0} : Set ℕ) (Set.Ioi 0)
(by simp : Disjoint ({0} : Set ℕ) (Set.Ioi 0))
(fun y => y ⟨0, by simp⟩)
(fun y : Set.Ioi (0 : ℕ) → ℝ => fun j => y ⟨j + 1, show 0 < j + 1 from Nat.succ_pos j⟩)
(measurable_pi_apply _) (by fun_prop)
exact h
theorem integral_pathProduct_le {Ω : Type*} [MeasurableSpace Ω]
{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
(hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
(hv : ∀ i, δ ≤ v i) (n : ℕ)
{shift : ℕ → (ℕ → ℝ) → ℝ} (hs : ∀ i, Measurable (shift i)) :
(∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤
(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
have hq : 0 ≤ q := by
have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
dsimp [q]
nlinarith
change (∫ ω, pathProduct θ n shift (fun i => X i ω) ∂P) ≤ q ^ n
induction n generalizing X m v shift with
| zero => simp [pathProduct]
| succ n ih =>
have hi := ih (fun i => hX (i + 1)) (hind.precomp (g := fun i : ℕ => i + 1) (by intro i j hij; exact Nat.add_right_cancel hij))
(fun i => hlaw (i + 1)) (fun i => hv (i + 1)) (fun i => hs (i + 1))
have he := integral_shifted_attenuation_sq_mul_le_of_indepFun (hlaw 0)
(show AEMeasurable (fun ω j => X (j + 1) ω) P from
(Measurable.of_eval (fun index => hX (index + 1))).aemeasurable)
(head_indep_tail hX hind) (hs 0)
(measurable_pathProduct θ n (fun i => hs (i + 1)))
(integrable_pathProduct_comp (fun i => hX (i + 1)) hθ.le n (fun i => hs (i + 1)))
(pathProduct_nonneg θ n (fun i => shift (i + 1))) hθ hδ (hv 0)
calc
(∫ ω, pathProduct θ (n + 1) shift (fun i => X i ω) ∂P) =
∫ ω, responseAttenuation θ (X 0 ω + shift 0 (fun j => X (j + 1) ω)) ^ 2 *
pathProduct θ n (fun i => shift (i + 1)) (fun j => X (j + 1) ω) ∂P := by
apply integral_congr_ae
exact Filter.Eventually.of_forall fun ω => by
have hz : pathTail 0 (fun i => X i ω) = (fun j => X (j + 1) ω) := by
funext j
simp only [pathTail, Nat.zero_add, Nat.add_comm]
simpa only [hz] using pathProduct_succ θ n shift (fun i => X i ω)
_ ≤ q * ∫ ω, pathProduct θ n (fun i => shift (i + 1))
(fun j => X (j + 1) ω) ∂P := he
_ ≤ q * q ^ n := mul_le_mul_of_nonneg_left hi hq
_ = q ^ (n + 1) := (pow_succ' q n).symm
lemma measurable_pathProduct_parameter {Y : Type*} [MeasurableSpace Y]
(θ : ℝ) (n : ℕ) {shift : Y → ℕ → (ℕ → ℝ) → ℝ}
(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
Measurable (fun p : Y × (ℕ → ℝ) => pathProduct θ n (shift p.1) p.2) := by
unfold pathProduct
apply Finset.measurable_fun_prod
intro i hi
have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
exact ha.comp (((measurable_pi_apply i).comp measurable_snd).add
((hs i).comp (measurable_fst.prodMk ((measurable_pathTail i).comp measurable_snd))))
theorem integral_pathProduct_parameter_le
{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
(ν : Measure Y) [IsProbabilityMeasure ν]
{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ}
(hX : ∀ i, Measurable (X i)) (hind : iIndepFun X P)
{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
(hv : ∀ i, δ ≤ v i) (n : ℕ)
{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
(∫ p : Y × Ω, pathProduct θ n (shift p.1) (fun i => X i p.2) ∂ν.prod P) ≤
(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
let F : Y × Ω → ℝ := fun p => pathProduct θ n (shift p.1) (fun i => X i p.2)
have hFm : Measurable F := by
dsimp only [F, pathProduct]
apply Finset.measurable_fun_prod
intro i hi
have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
have hg : Measurable (fun p : Y × Ω => (p.1, pathTail i (fun j => X j p.2))) := by
apply Measurable.prodMk measurable_fst
apply Measurable.of_eval
intro j
exact (hX (i + 1 + j)).comp measurable_snd
exact ha.comp (((hX i).comp measurable_snd).add ((hs i).comp hg))
have hFi : Integrable F (ν.prod P) := by
apply (integrable_const (1 : ℝ)).mono' hFm.aestronglyMeasurable
exact Filter.Eventually.of_forall fun p => by
change ‖pathProduct θ n (shift p.1) (fun i => X i p.2)‖ ≤ 1
rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
exact pathProduct_le_one hθ.le _ _ _
let q := 1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
change (∫ p, F p ∂ν.prod P) ≤ q ^ n
rw [integral_prod F hFi]
calc
(∫ y, ∫ ω, F (y, ω) ∂P ∂ν) ≤ ∫ _y, q ^ n ∂ν := by
apply integral_mono hFi.integral_prod_left (integrable_const _)
intro y
have hsy (i : ℕ) : Measurable (shift y i) := by
exact (hs i).comp (show Measurable (fun x : ℕ → ℝ => (y, x)) from
measurable_const.prodMk measurable_id)
change (∫ ω, pathProduct θ n (shift y) (fun i => X i ω) ∂P) ≤ q ^ n
exact integral_pathProduct_le (X := X) (m := m) (v := v) hX hind hlaw hθ hδ hv n
(shift := shift y) hsy
_ = q ^ n := by simp
theorem exists_uniform_path_attenuation_gap
{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ) :
∃ κ : ℝ, 0 < κ ∧ κ < 1 ∧
∀ (n : ℕ) (m : ℕ → ℝ) (v : ℕ → ℝ≥0), (∀ i, δ ≤ v i) →
∀ (shift : ℕ → (ℕ → ℝ) → ℝ), (∀ i, Measurable (shift i)) →
(∫ x, pathProduct θ n shift x
∂Measure.infinitePi (fun i => gaussianReal (m i) (v i))) ≤ (1 - κ) ^ n := by
let κ := ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
have ht : 0 < Real.tanh (gaussianSmallRadius δ) := by
rw [Real.tanh_eq_sinh_div_cosh]
exact div_pos (Real.sinh_pos_iff.mpr (gaussianSmallRadius_pos hδ)) (Real.cosh_pos _)
refine ⟨κ, by dsimp [κ]; positivity, ?_, ?_⟩
· have hp := mul_nonneg (sq_nonneg θ) (sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
have hlt := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
dsimp [κ]
nlinarith
· intro n m v hv shift hs
apply integral_pathProduct_le (m := m) (v := v) (fun i => measurable_pi_apply i)
(iIndepFun_infinitePi (fun _ => measurable_id)) _ hθ hδ hv n hs
intro i
exact ⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩
theorem integral_pathProduct_environment_le
{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
(henv : IndepFun Z (fun ω i => X i ω) P)
{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
(hv : ∀ i, δ ≤ v i) (n : ℕ)
{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
(∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) ≤
(1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2) ^ n := by
let μ := Measure.infinitePi (fun i => gaussianReal (m i) (v i))
have hXm : Measurable (fun ω i => X i ω) := Measurable.of_eval hX
have harray : HasLaw (fun ω i => X i ω) μ P :=
hind.hasLaw_infinitePi hlaw hXm.aemeasurable
have hpair : HasLaw (fun ω => (Z ω, fun i => X i ω)) ((P.map Z).prod μ) P := by
refine ⟨(hZ.prodMk hXm).aemeasurable, ?_⟩
rw [henv.map_prod_eq_prod_map_map hZ.aemeasurable hXm.aemeasurable, harray.map_eq]
have heval (i : ℕ) : HasLaw (fun x : ℕ → ℝ => x i) (gaussianReal (m i) (v i)) μ :=
⟨(measurable_pi_apply i).aemeasurable, Measure.infinitePi_map_eval _ i⟩
have hb := integral_pathProduct_parameter_le (P := μ) (P.map Z)
(fun i => measurable_pi_apply i) (iIndepFun_infinitePi (fun _ => measurable_id))
heval hθ hδ hv n (shift := shift) hs
calc
(∫ ω, pathProduct θ n (shift (Z ω)) (fun i => X i ω) ∂P) =
∫ p : Y × (ℕ → ℝ), pathProduct θ n (shift p.1) p.2 ∂(P.map Z).prod μ :=
hpair.integral_comp (measurable_pathProduct_parameter θ n hs).aestronglyMeasurable
_ ≤ _ := hb
def pathResponse (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
(x : ℕ → ℝ) : ℝ :=
θ ^ n * ∏ i ∈ Finset.range n, responseAttenuation θ (x i + shift i (pathTail i x))
lemma pathResponse_sq (θ : ℝ) (n : ℕ) (shift : ℕ → (ℕ → ℝ) → ℝ)
(x : ℕ → ℝ) : pathResponse θ n shift x ^ 2 = (θ ^ 2) ^ n * pathProduct θ n shift x := by
simp only [pathResponse, pathProduct, mul_pow, Finset.prod_pow]
rw [← pow_mul, ← pow_mul, Nat.mul_comm n 2]
theorem integral_pathResponse_sq_environment_le
{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
{P : Measure Ω} [IsProbabilityMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z) (hind : iIndepFun X P)
(henv : IndepFun Z (fun ω i => X i ω) P)
{m : ℕ → ℝ} {v : ℕ → ℝ≥0}
(hlaw : ∀ i, HasLaw (X i) (gaussianReal (m i) (v i)) P)
{θ : ℝ} (hθ : θ ^ 2 < 1) {δ : ℝ≥0} (hδ : 0 < δ)
(hv : ∀ i, δ ≤ v i) (n : ℕ)
{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
(∫ ω, pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2 ∂P) ≤
(θ ^ 2 * (1 - ((1 - θ ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2)) ^ n := by
simp_rw [pathResponse_sq]
rw [integral_const_mul, mul_pow]
exact mul_le_mul_of_nonneg_left
(integral_pathProduct_environment_le (X := X) (Z := Z) hX hZ hind henv hlaw hθ hδ hv n hs)
(pow_nonneg (sq_nonneg θ) n)
lemma integrable_pathResponse_sq_environment
{Y Ω : Type*} [MeasurableSpace Y] [MeasurableSpace Ω]
{P : Measure Ω} [IsFiniteMeasure P] {X : ℕ → Ω → ℝ} {Z : Ω → Y}
(hX : ∀ i, Measurable (X i)) (hZ : Measurable Z)
{θ : ℝ} (hθ : θ ^ 2 ≤ 1) (n : ℕ)
{shift : Y → ℕ → (ℕ → ℝ) → ℝ}
(hs : ∀ i, Measurable (fun p : Y × (ℕ → ℝ) => shift p.1 i p.2)) :
Integrable (fun ω => pathResponse θ n (shift (Z ω)) (fun i => X i ω) ^ 2) P := by
simp_rw [pathResponse_sq]
apply Integrable.const_mul
have hm : Measurable (fun ω => pathProduct θ n (shift (Z ω)) (fun i => X i ω)) := by
unfold pathProduct
apply Finset.measurable_fun_prod
intro i hi
have ha : Measurable (fun z : ℝ => responseAttenuation θ z ^ 2) := by
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
have hg : Measurable (fun ω => (Z ω, pathTail i (fun j => X j ω))) := by
apply Measurable.prodMk hZ
exact Measurable.of_eval (fun index => hX (i + 1 + index))
exact ha.comp ((hX i).add ((hs i).comp hg))
apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
exact Filter.Eventually.of_forall fun ω => by
rw [Real.norm_eq_abs, abs_of_nonneg (pathProduct_nonneg _ _ _ _)]
exact pathProduct_le_one hθ _ _ _
end Problem367.GaussianAttenuation
end
end OAI
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