| 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 | |