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135052a | 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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.Probability.SKBarriers.Scalar.ScalarInstability
import OAI.Probability.SKBarriers.SpinGlass.FiniteScalarGap
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem finiteParisiInf_temperature_upper {β γ δ : ℝ} (hβ : 0 < β) (hβγ : β ≤ γ)
(hδ : 0 < δ) (hgap : finiteParisiInf β ≤ Real.log 2+β^2/4-2*δ) :
finiteParisiInf γ ≤ finiteParisiInf β+
(γ^2-β^2)/4*(1-(β^2/γ^2)*(2*δ/β^2)^2) := by
have hγ : 0 < γ := hβ.trans_le hβγ
have hK : 0 ≤ (γ^2-β^2)/4 := by nlinarith
apply le_of_forall_pos_le_add
intro ε hε
have hsmall : 0 < min ε δ := lt_min hε hδ
obtain ⟨p,hp,hpLt⟩ := exists_lt_of_csInf_lt (finiteParisiValues_nonempty β)
(show finiteParisiInf β < finiteParisiInf β+min ε δ by linarith)
obtain ⟨k,Q,hm,hQ,rfl⟩ := hp
have hg : extendedQuantileParisi k β Q ≤ Real.log 2+β^2/4-δ := by
linarith [min_le_right ε δ]
have hS := quantileSecondMoment_of_gap hβ hδ.le Q hm hQ hg
obtain ⟨hm',hQ'⟩ := rescaledQuantiles_admissible hβ hβγ Q hm hQ
have HT := finiteParisiInf_le γ ((β/γ)^2 • Q) hm' hQ'
rw [extendedQuantileParisi_temperature_rescale hβ hγ Q] at HT
have HP : (γ^2-β^2)/4*(1-(β^2/γ^2)*quantileSecondMoment k Q) ≤
(γ^2-β^2)/4*(1-(β^2/γ^2)*(2*δ/β^2)^2) := by
apply mul_le_mul_of_nonneg_left _ hK
exact sub_le_sub_left (mul_le_mul_of_nonneg_left hS (by positivity)) 1
linarith [min_le_left ε δ]
theorem exists_strict_finiteParisiInf_secant {β : ℝ} (hβ : 1 < β) :
∃ γ : ℝ, β < γ ∧ (finiteParisiInf γ-finiteParisiInf β)/(γ-β) < β/2 := by
have hβpos : 0 < β := lt_trans zero_lt_one hβ
let δ := (Real.log 2+β^2/4-finiteParisiInf β)/2
have hδ : 0 < δ := by dsimp only [δ]; linarith [finiteParisiInf_lt_annealed hβ]
let c := (2*δ/β^2)^2
have hc : 0 < c := sq_pos_of_pos (div_pos (mul_pos (by norm_num) hδ) (sq_pos_of_pos hβpos))
let S := fun γ : ℝ => (γ+β)/4*(1-(β^2/γ^2)*c)
have HS : ContinuousAt S β := by
dsimp only [S]
fun_prop (disch := positivity)
have hS : S β < β/2 := by
dsimp only [S]
rw [div_self (sq_pos_of_pos hβpos).ne',one_mul]
nlinarith [mul_pos hβpos hc]
have HE : ∀ᶠ γ in 𝓝[>] β, S γ < β/2 :=
(HS.tendsto.eventually (eventually_lt_nhds hS)).filter_mono nhdsWithin_le_nhds
have HM : ∀ᶠ γ in 𝓝[>] β, β < γ := self_mem_nhdsWithin
obtain ⟨γ,hγβ,hγS⟩ := (HM.and HE).exists
have hβγ : β < γ := hγβ
have HU := finiteParisiInf_temperature_upper hβpos hβγ.le hδ
(show finiteParisiInf β ≤ Real.log 2+β^2/4-2*δ by dsimp only [δ]; linarith)
have hd : 0 < γ-β := sub_pos.mpr hβγ
have he : ((γ^2-β^2)/4*(1-(β^2/γ^2)*c))/(γ-β) = S γ := by
dsimp only [S]
field_simp [hd.ne', (hβpos.trans hβγ).ne']
ring
refine ⟨γ,hβγ,lt_of_le_of_lt ?_ hγS⟩
rw [← he]
exact div_le_div_of_nonneg_right (by dsimp only [c]; linarith [HU]) hd.le
end SK.Analytic
end
end
end
end OAI
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