File size: 4,041 Bytes
f61e547 | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.Probability.FreeIsing.Critical.NoisePathConstants
import OAI.Probability.FreeIsing.Ising.CopySmallLogLoss
import OAI.Probability.FreeIsing.Ising.NoiseTwoTails
import OAI.Probability.FreeIsing.Ising.OrientedStemTail
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingCopyResponseRowPaths
open IsingObservationInnovation IsingObservationFactor IsingCopySpectral
open IsingOrientedPath IsingCopyPathBounds IsingCopySmallLogLoss TreeKoopman
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
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
end Problem367.IsingCopyResponseRowPaths
end
end OAI
|