| import OAI.Probability.FreeIsing.Critical.NoisePathConstants |
| import OAI.Probability.FreeIsing.Ising.CopySmallLogLoss |
| import OAI.Probability.FreeIsing.Ising.NoiseTwoTails |
| import OAI.Probability.FreeIsing.Ising.OrientedStemTail |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| namespace Problem367.IsingCopyResponseRowPaths |
| open IsingObservationInnovation IsingObservationFactor IsingCopySpectral |
| open IsingOrientedPath IsingCopyPathBounds IsingCopySmallLogLoss TreeKoopman |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| theorem exists_noise_energy_long_constants |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) (T : ℝ≥0) : |
| ∃ B E c a q C₀ : ℝ, 0 ≤ B ∧ 0 ≤ E ∧ 0 < c ∧ 0 < a ∧ 1 < q ∧ 0 < C₀ ∧ |
| ∀ (t : ℝ≥0), 0 < t → t ≤ T → ∀ (u w : TreeVertex d), treeAdj u w → |
| ∀ v : TreeVertex d, C₀ / min (Real.sqrt (t : ℝ)) 1 ≤ (v.1.length : ℝ) → |
| noiseEnergy (μ := μ) hb t v ≤ |
| B * (logVariance hb μ u w t) * |
| ((v.1.length : ℝ) * Real.exp (-(c * min (Real.sqrt (t : ℝ)) 1) * v.1.length)) + |
| E * ((v.1.length : ℝ) ^ 2 * Real.exp (-a * q ^ v.1.length)) := by |
| have hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1) := by |
| apply (tanh_threshold_iff_branching_sq hd hb).mpr |
| simpa only [Nat.cast_sub (by omega : 1 ≤ d), Nat.cast_one] using hcritical.le |
| obtain ⟨C, hC, hraw⟩ := exists_noiseEnergy_bound_of_two_tails hμ hd hb hthreshold |
| obtain ⟨hη, hlengthC, hbase, hdecay⟩ := criticalNoise_constants_positive hd hb hcritical |
| let K := Real.exp (4 * beta / (1 - Real.tanh beta)) |
| refine ⟨4 * K * C, 4 * edgeLogBound d beta T ^ 2, |
| criticalNoiseEta d beta / 4, 1 / (2 * ((d : ℝ) - 1)), criticalNoiseTailBase d, |
| criticalNoiseLengthConstant beta, by positivity, by positivity, |
| div_pos hη (by norm_num), hdecay, hbase, hlengthC, ?_⟩ |
| intro t ht htT u w huw v hn |
| let n := v.1.length |
| 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 |
| obtain ⟨hn8, hlength⟩ := criticalNoise_length_admissible hd hb hcritical hω |
| (min_le_right _ _) hn |
| have htail (inward : Bool) (stem : List (Fin d)) (hstem : stem.IsChain (· ≠ ·)) |
| (m : ℕ) (hm : n / 2 ≤ m) : |
| (observationLaw μ) {x | K * |
| Real.exp (-((criticalNoiseEta d beta / 4) * ω) * n) < |
| extensionAverage hb t inward stem m x} ≤ |
| ENNReal.ofReal (Real.exp (-(1 / (2 * ((d : ℝ) - 1))) * criticalNoiseTailBase d ^ n)) := by |
| obtain ⟨hsum, hlevel, htail⟩ := half_path_cut_admissible hn8 hm |
| have hlevel' : ((n : ℝ) / 4) / 2 - 1 ≤ (m / 2 - 1 + 1 : ℕ) := by |
| convert hlevel using 1 |
| ring |
| have htail' : ((n : ℝ) / 4) / 2 ≤ (m - m / 2 : ℕ) := by |
| convert htail using 1 |
| ring |
| have h := measure_extensionAverage_gt_le_double_exp_common hμ hb hd hcritical ht |
| inward stem hstem (m / 2 - 1) (m - m / 2) hη.le |
| (show 0 ≤ (n : ℝ) / 4 by positivity) |
| (min_le_left _ _) (min_le_right _ _) hlevel' htail' hlength |
| simpa only [hsum, criticalNoise_threshold_eq, criticalNoise_tail_eq hd] using h |
| have hv : v.1.IsChain (· ≠ ·) := v.2 |
| have hvr : (reverseVertex v).1.IsChain (· ≠ ·) := (reverseVertex v).2 |
| have h := hraw t T htT u w huw v (n / 2) (by dsimp [n]; omega) |
| (K * Real.exp (-((criticalNoiseEta d beta / 4) * ω) * n)) |
| (Real.exp (-(1 / (2 * ((d : ℝ) - 1))) * criticalNoiseTailBase d ^ n)) |
| (by positivity) (Real.exp_pos _).le |
| (htail false (v.1.take (n / 2)) (hv.take _) (n - n / 2) (by omega)) |
| (htail true ((reverseVertex v).1.take (n - n / 2)) (hvr.take _) (n / 2) le_rfl) |
| convert h using 1 |
| simp only [logVariance, cavityLogDifference, n, ω] |
| ring |
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| end Problem367.IsingCopyResponseRowPaths |
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| end |
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| end OAI |
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