| import OAI.Probability.FreeIsing.Ising.CopyResponseRowPaths |
| import OAI.Probability.FreeIsing.Ising.RootDoubleExponential |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| namespace Problem367.IsingCopyResponseRowPaths |
| open IsingObservationFactor IsingConditionalCopyBounds ConnectedHull |
| open IsingRootAttenuation IsingRootDoubleExponential TreeOrbitAverage |
| open IsingCopyWeightedCut |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| theorem linearEnergy_eq_root_weight (hb : 0 ≤ beta) (t : ℝ≥0) (v : TreeVertex d) : |
| linearEnergy (μ := μ) hb t v = |
| ∫ z, (fieldDifference beta (root d) t z) ^ 2 * |
| rootAttenuation hb t v (observationCopy true z) ^ 2 ∂copies beta μ := rfl |
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| theorem linearEnergy_le_double_exp |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) (hn : 2 ≤ v.1.length) |
| {η : ℝ} (hη0 : 0 ≤ η) |
| (hη : η ≤ ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) / 4) |
| (hηlog : η ≤ Real.log ((d : ℝ) - 1) / 8) |
| (hlength : 8 * Real.log 2 ≤ |
| ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) * |
| (min (Real.sqrt (t : ℝ)) 1 * v.1.length)) : |
| linearEnergy (μ := μ) hb t v ≤ |
| (∫ z, (fieldDifference beta (root d) t z) ^ 2 ∂copies beta μ) * |
| Real.exp (-(η * min (Real.sqrt (t : ℝ)) 1) * v.1.length) + |
| (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2 * |
| Real.exp (-(((d : ℝ) - 1) ^ ((v.1.length : ℝ) / 4)) / |
| (2 * ((d : ℝ) - 1))) := by |
| let n := v.1.length |
| let k := n / 2 - 1 |
| let L := n - n / 2 |
| have hk : k + 1 = n / 2 := by dsimp [k, n]; omega |
| have hsum : k + 1 + L = n := by dsimp [L]; omega |
| let s : Extension ([] : List (Fin d)) (k + 1 + L) := |
| ⟨v.1, hsum.symm, by change v.1.IsChain (· ≠ ·); exact v.2⟩ |
| have hs : endpoint s = v := by apply Subtype.ext; rfl |
| have hlevel : (n : ℝ) / 2 - 1 ≤ (k + 1 : ℕ) := by |
| rw [hk] |
| have hround : n ≤ 2 * (n / 2) + 1 := by omega |
| have hr : (n : ℝ) ≤ 2 * (n / 2 : ℕ) + 1 := by exact_mod_cast hround |
| linarith |
| have htail : (n : ℝ) / 2 ≤ (L : ℝ) := by |
| have hround : n ≤ 2 * L := by dsimp [L]; omega |
| have hr : (n : ℝ) ≤ 2 * (L : ℝ) := by exact_mod_cast hround |
| linarith |
| have h := integral_root_weight_mul_sq_le_double_exp hμ hb hd hcritical ht true k L s |
| hη0 (Nat.cast_nonneg n) hη hηlog hlevel htail hlength |
| rw [hs, ← linearEnergy_eq_root_weight] at h |
| simpa only [mul_comm (Real.exp _) (∫ z, (fieldDifference beta (root d) t z) ^ 2 |
| ∂copies beta μ)] using h |
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|
| theorem exists_linear_energy_long_constants |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) : |
| ∃ c a q C₀ : ℝ, 0 < c ∧ 0 < a ∧ 1 < q ∧ 0 < C₀ ∧ |
| ∀ (T t : ℝ≥0), 0 < t → t ≤ T → ∀ v : TreeVertex d, |
| C₀ / min (Real.sqrt (t : ℝ)) 1 ≤ (v.1.length : ℝ) → |
| linearEnergy (μ := μ) hb t v ≤ |
| (∫ z, (fieldDifference beta (root d) t z) ^ 2 ∂copies beta μ) * |
| Real.exp (-(c * min (Real.sqrt (t : ℝ)) 1) * v.1.length) + |
| (2 * (T : ℝ) + 2 * (d : ℝ) * beta) ^ 2 * |
| Real.exp (-a * q ^ v.1.length) := by |
| let b : ℝ := (d : ℝ) - 1 |
| have hb1 : 1 < b := by |
| have hdR : (3 : ℝ) ≤ d := by exact_mod_cast hd |
| dsimp [b] |
| linarith |
| have hb0 : 0 < b := lt_trans zero_lt_one hb1 |
| have hθ : 0 < Real.tanh beta := by |
| by_contra h |
| have hz := le_antisymm (not_lt.mp h) (tanh_nonneg_of_beta_nonneg hb) |
| simp [hz] at hcritical |
| let γ := ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) |
| have hγ : 0 < γ := ComponentSignalAttenuation.lossCoefficient_pos hθ (Real.tanh_lt_one beta) |
| let c := min (γ / 4) (Real.log b / 8) |
| let a := 1 / (2 * b) |
| let q := b ^ (1 / 4 : ℝ) |
| let C₀ := max 2 (8 * Real.log 2 / γ) |
| have hc : 0 < c := lt_min (div_pos hγ (by norm_num)) |
| (div_pos (Real.log_pos hb1) (by norm_num)) |
| have ha : 0 < a := by dsimp [a]; positivity |
| have hq : 1 < q := Real.one_lt_rpow hb1 (by norm_num) |
| have hC₀ : 0 < C₀ := lt_of_lt_of_le (by norm_num) (le_max_left _ _) |
| refine ⟨c, a, q, C₀, hc, ha, hq, hC₀, ?_⟩ |
| intro T t ht htT v hn |
| let ω := min (Real.sqrt (t : ℝ)) 1 |
| have hω : 0 < ω := lt_min (Real.sqrt_pos.2 (show 0 < (t : ℝ) from ht)) zero_lt_one |
| have hω1 : ω ≤ 1 := min_le_right _ _ |
| have hlarge : C₀ ≤ (v.1.length : ℝ) * ω := (div_le_iff₀ hω).mp hn |
| have hnR : (2 : ℝ) ≤ v.1.length := by |
| have hmul := mul_le_mul_of_nonneg_left hω1 (Nat.cast_nonneg v.1.length : (0 : ℝ) ≤ _) |
| have htwo : (2 : ℝ) ≤ C₀ := le_max_left _ _ |
| nlinarith |
| have hn2 : 2 ≤ v.1.length := by exact_mod_cast hnR |
| have hlarge' : 8 * Real.log 2 / γ ≤ ω * (v.1.length : ℝ) := |
| (le_max_right _ _).trans (by simpa only [mul_comm] using hlarge) |
| have hlength : 8 * Real.log 2 ≤ γ * (ω * v.1.length) := by |
| have h := (div_le_iff₀ hγ).mp hlarge' |
| nlinarith |
| have h := linearEnergy_le_double_exp hμ hb hd hcritical ht v hn2 hc.le |
| (min_le_left _ _) (min_le_right _ _) hlength |
| have hpow : b ^ ((v.1.length : ℝ) / 4) = q ^ v.1.length := by |
| rw [show (v.1.length : ℝ) / 4 = (1 / 4 : ℝ) * v.1.length by ring, |
| Real.rpow_mul hb0.le, Real.rpow_natCast] |
| have hexp : -b ^ ((v.1.length : ℝ) / 4) / (2 * b) = -a * q ^ v.1.length := by |
| rw [hpow] |
| dsimp [a] |
| ring |
| change linearEnergy (μ := μ) hb t v ≤ _ + |
| (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2 * |
| Real.exp (-b ^ ((v.1.length : ℝ) / 4) / (2 * b)) at h |
| rw [hexp] at h |
| apply h.trans |
| apply add_le_add le_rfl |
| apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le |
| apply pow_le_pow_left₀ (by positivity) |
| have htTR : (t : ℝ) ≤ T := htT |
| linarith |
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| end Problem367.IsingCopyResponseRowPaths |
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| end |
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| end OAI |
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