| import OAI.Probability.InvariantIsing.Fields.FieldBounds |
| import OAI.Probability.InvariantIsing.Spectral.SpectralEnergy |
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|
| namespace OAI |
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| /-! Finiteness and finite-entropy approximate minimizers of the finite-spectrum |
| variational problem. These justify taking the infimum in the pressure bound. -/ |
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| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators |
|
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| namespace InvariantIsing |
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|
| theorem finiteSpectralFunctional_abs_le {ι : Type*} [Fintype ι] |
| (ρ eig : ι → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) |
| (K : ℝ) (heig : ∀ a, |eig a| ≤ K) (p : OverlapPath) : |
| |spectralFunctional (finiteR ρ eig hρ hρsum) p| ≤ K / 2 := by |
| have hb x : |finiteR ρ eig hρ hρsum x - 0| ≤ K := by |
| simpa only [sub_zero] using abs_le.mpr (finiteR_mem_interval ρ eig hρ hρsum |
| (fun a => (abs_le.mp (heig a)).1) (fun a => (abs_le.mp (heig a)).2) x) |
| simpa only [spectralFunctional_constant, zero_div, sub_zero] using |
| abs_spectralFunctional_sub_le (finiteR ρ eig hρ hρsum) (fun _ => 0) |
| (continuous_finiteR ρ eig hρ hρsum) continuous_const K (fun x _ => hb x) p |
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|
| theorem finiteVariational_mem_interval {ι : Type*} [Fintype ι] |
| (ρ eig : ι → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) |
| (K : ℝ) (heig : ∀ a, |eig a| ≤ K) : |
| (-(K / 2) : ℝ) ≤ variationalFunctional (finiteR ρ eig hρ hρsum) ∧ |
| variationalFunctional (finiteR ρ eig hρ hρsum) ≤ (K / 2 : ℝ) := by |
| constructor |
| · apply le_iInf |
| intro p |
| have hs := (abs_le.mp (finiteSpectralFunctional_abs_le ρ eig hρ hρsum K heig p)).1 |
| have he : ((-(K / 2) : ℝ) : EReal) ≤ |
| (spectralFunctional (finiteR ρ eig hρ hρsum) p : EReal) := by exact_mod_cast hs |
| have hadd := add_le_add (entropyFunctional_nonneg p) |
| (le_rfl : (spectralFunctional (finiteR ρ eig hρ hρsum) p : EReal) ≤ _) |
| exact he.trans (by simpa only [zero_add] using hadd) |
| · let p := constantPath 0 (by constructor <;> norm_num) |
| have hi := iInf_le (fun p : OverlapPath => entropyFunctional p + |
| (spectralFunctional (finiteR ρ eig hρ hρsum) p : EReal)) p |
| have hs := (abs_le.mp (finiteSpectralFunctional_abs_le ρ eig hρ hρsum K heig p)).2 |
| simp only [p, entropyFunctional_zero, zero_add] at hi |
| exact hi.trans (by exact_mod_cast hs) |
|
|
| theorem finiteVariational_ne_top_bot {ι : Type*} [Fintype ι] |
| (ρ eig : ι → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) : |
| variationalFunctional (finiteR ρ eig hρ hρsum) ≠ ⊤ ∧ |
| variationalFunctional (finiteR ρ eig hρ hρsum) ≠ ⊥ := by |
| let K := ∑ a, |eig a| |
| have heig a : |eig a| ≤ K := Finset.single_le_sum |
| (fun a _ => abs_nonneg (eig a)) (Finset.mem_univ a) |
| have hb := finiteVariational_mem_interval ρ eig hρ hρsum K heig |
| exact ⟨ne_top_of_le_ne_top (EReal.coe_ne_top _) hb.2, |
| ne_bot_of_le_ne_bot (EReal.coe_ne_bot _) hb.1⟩ |
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|
| theorem exists_finiteEntropy_trial_near_finiteVariational {ι : Type*} [Fintype ι] |
| (ρ eig : ι → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ p : OverlapPath, ∃ S : ℝ, entropyFunctional p = (S : EReal) ∧ |
| S + spectralFunctional (finiteR ρ eig hρ hρsum) p < |
| (variationalFunctional (finiteR ρ eig hρ hρsum)).toReal + ε := by |
| have hf := finiteVariational_ne_top_bot ρ eig hρ hρsum |
| have hlt : variationalFunctional (finiteR ρ eig hρ hρsum) < |
| ((variationalFunctional (finiteR ρ eig hρ hρsum)).toReal + ε : ℝ) := by |
| rw [← EReal.coe_toReal hf.1 hf.2] |
| exact_mod_cast (show (variationalFunctional (finiteR ρ eig hρ hρsum)).toReal < |
| (variationalFunctional (finiteR ρ eig hρ hρsum)).toReal + ε by linarith) |
| obtain ⟨p, hp⟩ := iInf_lt_iff.mp hlt |
| have htop : entropyFunctional p ≠ ⊤ := by |
| intro he |
| simp only [he, EReal.top_add_coe, not_top_lt] at hp |
| have hbot : entropyFunctional p ≠ ⊥ := |
| ne_bot_of_le_ne_bot (by simp) (entropyFunctional_nonneg p) |
| refine ⟨p, (entropyFunctional p).toReal, (EReal.coe_toReal htop hbot).symm, ?_⟩ |
| rw [← EReal.coe_toReal htop hbot, ← EReal.coe_add] at hp |
| exact_mod_cast hp |
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|
| end InvariantIsing |
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| end |
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| end OAI |
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