| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.FiniteTemperatureBound |
| import OAI.Probability.SKBarriers.Coverage.EquilibriumEnergy |
|
|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| open scoped Topology |
|
|
| theorem equilibrium_energy_gap {β : ℝ} (hβ : 1 < β) : |
| ∃ e₀ : ℝ, e₀ < β/2 ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| equilibriumEnergy β N ≤ e₀ := by |
| obtain ⟨γ,hβγ,hsec⟩ := exists_strict_finiteParisiInf_secant hβ |
| obtain ⟨e₀,hleft,hright⟩ := exists_between hsec |
| have hd : 0 < γ-β := sub_pos.mpr hβγ |
| have HT : Tendsto (fun N : ℕ => (finiteParisiInf γ-quenchedPressure β N)/(γ-β)) |
| atTop (𝓝 ((finiteParisiInf γ-finiteParisiInf β)/(γ-β))) := |
| (tendsto_const_nhds.sub (quenchedPressure_tendsto_finiteParisiInf |
| (lt_trans zero_lt_one hβ))).div_const (γ-β) |
| have HE := HT.eventually (eventually_lt_nhds hleft) |
| obtain ⟨N₀,hN₀⟩ := eventually_atTop.mp HE |
| refine ⟨e₀,hright,max N₀ 1,?_⟩ |
| intro N hN |
| have hNpos : 0 < N := lt_of_lt_of_le Nat.zero_lt_one ((le_max_right N₀ 1).trans hN) |
| have HU := equilibriumEnergy_le_secant hNpos hβγ |
| have HQ := quenched_le_finiteParisiInf hNpos γ |
| have HD := div_le_div_of_nonneg_right (sub_le_sub_right HQ (quenchedPressure β N)) hd.le |
| exact (HU.trans HD).trans (hN₀ N ((le_max_left N₀ 1).trans hN)).le |
|
|
| theorem pressure_derivative_energy_gap {β : ℝ} (hβ : 1 < β) : |
| ∃ e₀ : ℝ, e₀ < β/2 ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| HasDerivAt (fun γ => quenchedPressure γ N) (equilibriumEnergy β N) β ∧ |
| equilibriumEnergy β N ≤ e₀ := by |
| obtain ⟨e₀,he₀,N₀,hN₀⟩ := equilibrium_energy_gap hβ |
| refine ⟨e₀,he₀,max N₀ 1,?_⟩ |
| intro N hN |
| exact ⟨quenchedPressure_hasDerivAt (lt_of_lt_of_le Nat.zero_lt_one |
| ((le_max_right N₀ 1).trans hN)) β,hN₀ N ((le_max_left N₀ 1).trans hN)⟩ |
| end SK.Analytic |
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|
| end |
| end |
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| end |
|
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| end OAI |
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