| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.ThermodynamicLimit |
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| 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 |
|
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| def gibbsEnergyTail {N : ℕ} (β e : ℝ) (J : Disorder N) : ℝ := |
| ∑ s : Config N, if (N:ℝ)*e < hamiltonian J s then gibbs β J s else 0 |
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| theorem gibbsEnergyTail_le {N : ℕ} {β γ : ℝ} (hβγ : β < γ) |
| (e : ℝ) (J : Disorder N) : |
| gibbsEnergyTail β e J ≤ |
| Real.exp (logPartition γ J-logPartition β J-(γ-β)*(N:ℝ)*e) := by |
| have hZβ := partition_pos β J |
| have hZγ := partition_pos γ J |
| have he : Real.exp (logPartition γ J-logPartition β J-(γ-β)*(N:ℝ)*e) = |
| Real.exp (-((γ-β)*(N:ℝ)*e))*partition γ J/partition β J := by |
| rw [show logPartition γ J-logPartition β J-(γ-β)*(N:ℝ)*e = |
| -((γ-β)*(N:ℝ)*e)+(logPartition γ J-logPartition β J) by ring, |
| Real.exp_add,Real.exp_sub] |
| simp only [logPartition,Real.exp_log hZγ,Real.exp_log hZβ] |
| ring |
| rw [he] |
| unfold gibbsEnergyTail |
| calc |
| _ ≤ ∑ s : Config N, |
| Real.exp (-((γ-β)*(N:ℝ)*e))*Real.exp (γ*hamiltonian J s)/partition β J := by |
| apply Finset.sum_le_sum |
| intro s _ |
| split_ifs with hs |
| · unfold gibbs |
| apply div_le_div_of_nonneg_right _ hZβ.le |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have H := mul_le_mul_of_nonneg_left hs.le (sub_pos.mpr hβγ).le |
| nlinarith |
| · positivity |
| _ = _ := by rw [← Finset.sum_div,← Finset.mul_sum]; rfl |
|
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| theorem logPartition_upper_tail_linear {N : ℕ} (hN : 0 < N) {β a : ℝ} |
| (hβ : β ≠ 0) (ha : 0 ≤ a) : |
| (disorderLaw N).real {J | a*(N:ℝ) ≤ logPartition β J- |
| ∫ K, logPartition β K ∂disorderLaw N} ≤ |
| Real.exp (-(4*a^2/(Real.pi^2*β^2))*(N:ℝ)) := by |
| have H := logPartition_upper_tail hN β (a*(N:ℝ)) (by positivity) |
| convert H using 1 |
| congr 1 |
| change -(4*a^2/(Real.pi^2*β^2))*(N:ℝ) = |
| -(a*(N:ℝ))^2/(2*(Real.pi^2*β^2*(N:ℝ)/8)) |
| have hn : (N:ℝ) ≠ 0 := by exact_mod_cast hN.ne' |
| field_simp |
| ring |
|
|
| theorem logPartition_lower_tail_linear {N : ℕ} (hN : 0 < N) {β a : ℝ} |
| (hβ : β ≠ 0) (ha : 0 ≤ a) : |
| (disorderLaw N).real {J | a*(N:ℝ) ≤ |
| (∫ K, logPartition β K ∂disorderLaw N)-logPartition β J} ≤ |
| Real.exp (-(4*a^2/(Real.pi^2*β^2))*(N:ℝ)) := by |
| have H := logPartition_lower_tail hN β (a*(N:ℝ)) (by positivity) |
| convert H using 1 |
| congr 1 |
| change -(4*a^2/(Real.pi^2*β^2))*(N:ℝ) = |
| -(a*(N:ℝ))^2/(2*(Real.pi^2*β^2*(N:ℝ)/8)) |
| have hn : (N:ℝ) ≠ 0 := by exact_mod_cast hN.ne' |
| field_simp |
| ring |
|
|
| theorem gibbsEnergyTail_probability_le {N : ℕ} (hN : 0 < N) {β γ e a : ℝ} |
| (hβ : 0 < β) (hβγ : β < γ) (ha : 0 < a) |
| (hP : quenchedPressure γ N-quenchedPressure β N ≤ (γ-β)*e-3*a) : |
| (disorderLaw N).real {J | Real.exp (-a*(N:ℝ)) < gibbsEnergyTail β e J} ≤ |
| 2*Real.exp (-(4*a^2/(Real.pi^2*γ^2))*(N:ℝ)) := by |
| let A := {J : Disorder N | a*(N:ℝ) ≤ logPartition γ J- |
| ∫ K, logPartition γ K ∂disorderLaw N} |
| let B := {J : Disorder N | a*(N:ℝ) ≤ |
| (∫ K, logPartition β K ∂disorderLaw N)-logPartition β J} |
| have hsub : {J : Disorder N | Real.exp (-a*(N:ℝ)) < gibbsEnergyTail β e J} ⊆ A ∪ B := by |
| intro J hJ |
| by_contra hbad |
| have hA : logPartition γ J-(∫ K, logPartition γ K ∂disorderLaw N) < a*(N:ℝ) := |
| lt_of_not_ge (fun h => hbad (Or.inl h)) |
| have hB : (∫ K, logPartition β K ∂disorderLaw N)-logPartition β J < a*(N:ℝ) := |
| lt_of_not_ge (fun h => hbad (Or.inr h)) |
| have hPlin := mul_le_mul_of_nonneg_right hP (show (0:ℝ) ≤ N by positivity) |
| have hNne : (N:ℝ) ≠ 0 := by exact_mod_cast hN.ne' |
| simp only [quenchedPressure,sub_mul,div_mul_cancel₀ _ hNne] at hPlin |
| have hbound := gibbsEnergyTail_le hβγ e J |
| have hsmall : logPartition γ J-logPartition β J-(γ-β)*(N:ℝ)*e ≤ -a*(N:ℝ) := by |
| nlinarith |
| exact (not_lt_of_ge (hbound.trans (Real.exp_le_exp.mpr hsmall))) hJ |
| have hγ : 0 < γ := hβ.trans hβγ |
| have Hγ := logPartition_upper_tail_linear hN hγ.ne' ha.le |
| have Hβ := logPartition_lower_tail_linear hN hβ.ne' ha.le |
| have hrate : 4*a^2/(Real.pi^2*γ^2) ≤ 4*a^2/(Real.pi^2*β^2) := by |
| apply div_le_div_of_nonneg_left (by positivity) (by positivity) |
| apply mul_le_mul_of_nonneg_left _ (sq_nonneg Real.pi) |
| nlinarith |
| have Hβ' : (disorderLaw N).real B ≤ |
| Real.exp (-(4*a^2/(Real.pi^2*γ^2))*(N:ℝ)) := by |
| apply Hβ.trans |
| apply Real.exp_le_exp.mpr |
| exact mul_le_mul_of_nonneg_right (neg_le_neg hrate) (by positivity) |
| exact (measureReal_mono hsub).trans ((measureReal_union_le A B).trans (by linarith)) |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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