| import Mathlib |
| import OAI.Probability.SKGap.Stability.PlantedFieldCenter |
| import OAI.Probability.SKGap.Terminal.GibbsEventProbability |
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| namespace OAI |
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| section |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory Metric Set |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Matrix |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| namespace SKGapCutoff.Regression |
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|
| lemma quenched_field_average_exponential (β : ℝ) (hβ : 0 < β) (hβ1 : β < 1) |
| (f : ℝ → ℝ) {K : ℝ≥0} (hK : 0 < K) (hf : LipschitzWith K f) |
| (B : ℝ) (hB : 0 < B) (hbound : ∀ z, |f z| ≤ B) |
| (ε : ℝ) (hε : 0 < ε) : |
| ∃ c : ℝ, 0 < c ∧ Tendsto (fun n => disorderLaw β n {g | |
| ENNReal.ofReal (Real.exp (-c*n)) ≤ |
| gibbsEventProbability (fun x : Spin n => {h : GaussianCoordinates n | |
| ε ≤ |(∑ i : Fin n, f (gaugedField x h i))/(n:ℝ) - |
| ∫ z, f (β^2+β*z) ∂gaussianReal 0 1|}) g}) atTop (nhds 0) := by |
| obtain ⟨c, hc, hs⟩ := planted_field_average_exponential β hβ f hK hf B hB hbound ε hε |
| refine ⟨c/2, by positivity, ?_⟩ |
| have hb : β^2 < 1 := by nlinarith |
| apply quenched_gibbs_events_exponential β hb c hc _ _ hs |
| intro n x |
| apply measurableSet_le measurable_const |
| apply Measurable.abs |
| apply Measurable.sub_const |
| apply Measurable.div_const |
| apply Finset.measurable_sum |
| intro i _ |
| exact hf.continuous.measurable.comp |
| ((measurable_pi_apply i).comp (continuous_gaugedField x).measurable) |
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| end SKGapCutoff.Regression |
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| open scoped BigOperators |
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| end |
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| end OAI |
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