| import OAI.MathematicalPhysics.CriticalSK.FunctionalMeasurability |
| import OAI.MathematicalPhysics.CriticalSK.MixingMeasurability |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Topology NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| namespace CriticalSK |
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| theorem critical_logSobolev_lower_bound (δ : ℝ) (hδ : 0 < δ) : |
| Tendsto (fun n => (disorderLaw n).real |
| {W | (n : ℝ) ^ (2 / 3 - δ) ≤ logSobolevConstant W}) atTop (𝓝 1) := by |
| have hp := critical_relaxation_lower_bound δ hδ |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| filter_upwards [(tendsto_order.mp hp).1 a ha, eventually_gt_atTop 0] with n hn hnpos |
| refine lt_of_lt_of_le hn (measureReal_mono ?_) |
| intro W hW |
| exact hW.trans (relaxationTime_le_logSobolevConstant W hnpos) |
| · intro a ha |
| exact Filter.Eventually.of_forall (fun _ => lt_of_le_of_lt measureReal_le_one ha) |
|
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| theorem critical_lower_exponents |
| (ε : ℝ) (hε : 0 < ε) (hεhalf : ε < 1 / 2) (δ : ℝ) (hδ : 0 < δ) : |
| Tendsto (fun n => (disorderLaw n).real |
| {W | (n : ℝ) ^ (2 / 3 - δ) ≤ continuousMixingAt W ε}) atTop (𝓝 1) ∧ |
| Tendsto (fun n => (disorderLaw n).real |
| {W | (n : ℝ) ^ (5 / 3 - δ) ≤ (discreteMixingAt W ε : ℝ)}) atTop (𝓝 1) ∧ |
| Tendsto (fun n => (disorderLaw n).real |
| {W | (n : ℝ) ^ (2 / 3 - δ) ≤ relaxationTime W}) atTop (𝓝 1) ∧ |
| Tendsto (fun n => (disorderLaw n).real |
| {W | (n : ℝ) ^ (2 / 3 - δ) ≤ logSobolevConstant W}) atTop (𝓝 1) := by |
| have hm := critical_mixing_lower_bounds_at ε hε hεhalf δ hδ |
| exact ⟨hm.1, hm.2, critical_relaxation_lower_bound δ hδ, |
| critical_logSobolev_lower_bound δ hδ⟩ |
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| end CriticalSK |
| end |
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| end OAI |
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