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c4cdbbd | 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 79 80 81 82 83 84 85 | import OAI.Probability.InvariantIsing.Fields.FieldBounds
import OAI.Probability.InvariantIsing.Spectral.SpectralEnergy
namespace OAI
/-! Finiteness and finite-entropy approximate minimizers of the finite-spectrum
variational problem. These justify taking the infimum in the pressure bound. -/
noncomputable section
open MeasureTheory Set
open scoped BigOperators
namespace InvariantIsing
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
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⟩
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
end InvariantIsing
end
end OAI
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