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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 | import OAI.Probability.FreeIsing.Estimates.RadialRepresentation
namespace OAI
noncomputable section
namespace Problem367
namespace RadialWalk
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
theorem norm_power_sq_eq_inner_double (A : H →L[ℝ] H)
(hA : (A : H →ₗ[ℝ] H).IsSymmetric) (x : H) (n : ℕ) :
‖(A ^ n) x‖ ^ 2 = inner ℝ x ((A ^ (2 * n)) x) := by
have hp : ((A ^ n : H →L[ℝ] H) : H →ₗ[ℝ] H).IsSymmetric := by
simpa only [ContinuousLinearMap.toLinearMap_pow] using hA.pow n
calc
‖(A ^ n) x‖ ^ 2 = inner ℝ ((A ^ n) x) ((A ^ n) x) :=
(real_inner_self_eq_norm_sq _).symm
_ = inner ℝ x ((A ^ n) ((A ^ n) x)) := hp x ((A ^ n) x)
_ = inner ℝ x ((A ^ (2 * n)) x) := by
rw [show 2 * n = n + n by omega, pow_add]
rfl
theorem norm_power_sq_le_of_radial_bound {d : ℕ}
(S : Fin d → H →L[ℝ] H) (hS : ∀ c, S c * S c = 1)
(hA : (operatorAverage S : H →ₗ[ℝ] H).IsSymmetric)
(hd : 3 ≤ d) (x : H) {C : ℝ} (hC : 0 ≤ C)
(hcov : ∀ v, |inner ℝ x (wordOperator S v x)| ≤ C * treeWeight d v)
(n : ℕ) :
‖(operatorAverage S ^ n) x‖ ^ 2 ≤ C * rate d ^ (2 * n) := by
rw [norm_power_sq_eq_inner_double _ hA]
exact (le_abs_self _).trans (matrix_coefficient_power_bound S hS hd x hC hcov (2 * n))
theorem norm_power_le_of_radial_bound {d : ℕ}
(S : Fin d → H →L[ℝ] H) (hS : ∀ c, S c * S c = 1)
(hA : (operatorAverage S : H →ₗ[ℝ] H).IsSymmetric)
(hd : 3 ≤ d) (x : H) {C : ℝ} (hC : 0 ≤ C)
(hcov : ∀ v, |inner ℝ x (wordOperator S v x)| ≤ C * treeWeight d v)
(n : ℕ) :
‖(operatorAverage S ^ n) x‖ ≤ Real.sqrt C * rate d ^ n := by
apply (sq_le_sq₀ (norm_nonneg _)
(mul_nonneg (Real.sqrt_nonneg _) (pow_nonneg (rate_nonneg d) n))).mp
calc
‖(operatorAverage S ^ n) x‖ ^ 2 ≤ C * rate d ^ (2 * n) :=
norm_power_sq_le_of_radial_bound S hS hA hd x hC hcov n
_ = (Real.sqrt C * rate d ^ n) ^ 2 := by
rw [mul_pow, Real.sq_sqrt hC, ← pow_mul]
rw [Nat.mul_comm n 2]
end RadialWalk
end Problem367
end
end OAI
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