| import OAI.Probability.FreeIsing.Ising.CopyCriticalSpatialBounds |
| import OAI.Probability.FreeIsing.Ising.CopyNoiseLongBound |
|
|
| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| open scoped NNReal |
| namespace Problem367.IsingCopyResponseRowPaths |
| open IsingObservationFactor IsingCopySmallLogLoss IsingCriticalSmallLogLoss ConnectedHull |
|
|
| theorem exists_critical_spatial_endpoint_bounds |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
| (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) : |
| ∃ M : ℝ, 1 ≤ M ∧ ∀ (t : ℝ≥0), 0 < t → t ≤ T → |
| ∀ (u w : TreeVertex d), treeAdj u w → 0 < siteVariance beta μ (root d) t → |
| CriticalSpatialEndpointBounds |
| (∫ z, IsingCopyEnergy.density beta (root d) t z ∂copies beta μ) |
| (∑' n, noiseShellEnergy (μ := μ) hb t n) |
| (∑' n, linearShellEnergy (μ := μ) hb t n) |
| (∑' n, Real.sqrt (noiseShellEnergy (μ := μ) hb t n)) |
| (logVariance hb μ u w t) (siteVariance beta μ (root d) t) |
| M (varianceScale d beta T) (min (Real.sqrt (t : ℝ)) 1) := by |
| obtain ⟨B, E, c, a, q, C₀, hB, hE, hc, ha, hq, _, hnoise⟩ := |
| exists_noise_energy_long_constants hμ hd hb hcritical T |
| exact exists_actual_spatial_endpoint_of_noise_bound hμ hd hb hcritical T |
| hB hE hc ha hq hnoise |
|
|
| end Problem367.IsingCopyResponseRowPaths |
|
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| end |
|
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| end OAI |
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|
