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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 | import OAI.Probability.FreeIsing.Ising.CopyCriticalSpatialBounds
import OAI.Probability.FreeIsing.Ising.CopyNoiseLongBound
namespace OAI
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
end
end OAI
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