| import Mathlib |
| import OAI.Probability.SKGap.Terminal.TerminalDeterministic |
| import OAI.Probability.SKGap.Brownian.GibbsContinuity |
| import OAI.Probability.SKGap.Localization.RareCoordinate |
| import OAI.Probability.SKGap.Posterior.GaussianChannel |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Real Set Matrix |
| open scoped BigOperators ENNReal RealInnerProductSpace Matrix.Norms.L2Operator |
| variable {n : ℕ} |
|
|
| def observationField (T : ℝ) (x : Spin n) (z : Fin n→ℝ) : Fin n→ℝ := |
| fun i=>T*spinValue (x i)+sqrt T*z i |
|
|
| lemma spinValue_sign (b : Bool) : spinValue b=1 ∨ spinValue b= -1 := by |
| cases b <;> simp |
|
|
| lemma rareCount_field (T : ℝ) (x : Spin n) (z : Fin n→ℝ) : |
| rareCount (fun i=>{u : ℝ | |T*spinValue (x i)+sqrt T*u| < T/2}) z= |
| ((Finset.univ.filter (fun i=>|observationField T x z i| < T/2)).card:ℝ) := by |
| classical |
| simp only [rareCount,observationField,Set.mem_ofPred_eq] |
| simp only [Finset.sum_boole] |
| rfl |
| |
| lemma observation_smallField_tail {T α : ℝ} (hT : 0 < T) |
| (hp : exp (-T/8)*(exp 1-1) ≤ α/4) (x : Spin n) : |
| (channelReference n).real {z | α*(n:ℝ)/2 < |
| ((Finset.univ.filter (fun i=>|observationField T x z i| < T/2)).card:ℝ)} ≤ |
| exp (-α*(n:ℝ)/4) := by |
| classical |
| have hh := rareCount_tail (fun _ : Fin n=>gaussianReal 0 1) |
| (fun i=>{u : ℝ | |T*spinValue (x i)+sqrt T*u| < T/2}) |
| (fun _=>isOpen_lt (by fun_prop) continuous_const |>.measurableSet) |
| (a:=α*(n:ℝ)/2) (fun i=>gaussian_observation_small hT (spinValue_sign (x i))) |
| simp only [rareCount_field,Fintype.card_fin] at hh |
| apply hh.trans |
| apply exp_le_exp.mpr |
| have hm := mul_le_mul_of_nonneg_left hp (Nat.cast_nonneg n : (0:ℝ) ≤ n) |
| nlinarith only [hm] |
|
|
| theorem terminal_probability_fixedSpin (g : Disorder n) {M T α ε R : ℝ} |
| (hM : 0 ≤ M) (hT : 0 < T) (hε : 0 ≤ ε) |
| (hJ : ‖Matrix.of (coupling g)‖ ≤ M) |
| (hU : ‖terminalU g‖ ≤ M) (hQ : ‖terminalQ g‖ ≤ M) |
| (hc : ∀ i j,|tanh (coupling g i j)| ≤ 1/1000) |
| (hR : ∀ i,∑ j,tanh (coupling g i j)^4 ≤ R) |
| (hsparse : ∀ S : Finset (Fin n),(S.card:ℝ) ≤ α*(n:ℝ) → |
| ∀ w : EuclideanSpace ℝ (Fin n),(∀ i,i ∉ S → w i=0) → |
| |⟪w,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) (terminalU g) w⟫| ≤ ε*‖w‖^2 ∧ |
| |⟪w,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) (terminalQ g) w⟫| ≤ ε*‖w‖^2) |
| (hsmall : M^2 ≤ (T/4)^2*α/2) |
| (hnum : 800000*R+400001*(ε+M*(2*sqrt (1/cosh (T/4)^2)+1/cosh (T/4)^2)) ≤ 1/2) |
| (hp : exp (-T/8)*(exp 1-1) ≤ α/4) (x : Spin n) : |
| (channelReference n).real {z | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T x z) f ≤ 2*dirichlet g (observationField T x z) f} ≤ |
| exp (-α*(n:ℝ)/4) := by |
| apply (measureReal_mono (μ:=channelReference n) ?_ (measure_ne_top _ _)).trans |
| (observation_smallField_tail hT hp x) |
| intro z hz |
| by_contra hn |
| change ¬α*(n:ℝ)/2 < ((Finset.univ.filter (fun i=>|observationField T x z i| < T/2)).card:ℝ) at hn |
| have hb := le_of_not_gt hn |
| apply hz |
| apply terminal_deterministic g (observationField T x z) hM (by positivity : 0 < T/4) hε hJ hU hQ hc hR hsparse |
| ?_ hsmall hnum |
| simpa only [show 2*(T/4)=T/2 by ring] using hb |
| |
| lemma terminal_bad_measurable (g : Disorder n) (T : ℝ) : |
| MeasurableSet {p : Spin n × (Fin n→ℝ) | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T p.1 p.2) f ≤ 2*dirichlet g (observationField T p.1 p.2) f} := by |
| apply ((terminal_goodSet_closed g 2).measurableSet.compl).preimage |
| unfold observationField |
| apply Measurable.of_eval |
| intro i |
| exact (measurable_const.mul ((measurable_of_finite (spinValue)).comp ((measurable_pi_apply i).comp measurable_fst))).add |
| (measurable_const.mul ((measurable_pi_apply i).comp measurable_snd)) |
| |
| lemma terminal_probability_anyLaw (g : Disorder n) {T c : ℝ} |
| (hh : ∀ x : Spin n,(channelReference n).real {z | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T x z) f ≤ 2*dirichlet g (observationField T x z) f} ≤ exp (-c*(n:ℝ))) |
| (μ : Measure (Spin n)) [IsProbabilityMeasure μ] : |
| (μ.prod (channelReference n)).real {p | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T p.1 p.2) f ≤ 2*dirichlet g (observationField T p.1 p.2) f} ≤ exp (-c*(n:ℝ)) := by |
| have hb (x : Spin n) : channelReference n {z | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T x z) f ≤ 2*dirichlet g (observationField T x z) f} ≤ ENNReal.ofReal (exp (-c*(n:ℝ))) := by |
| exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mp |
| (by simpa only [measureReal_def,ENNReal.toReal_ofReal (exp_pos _).le] using hh x) |
| have he := Measure.prod_apply (μ:=μ) (ν:=channelReference n) (terminal_bad_measurable g T) |
| have hh' : μ.prod (channelReference n) {p | ¬∀ f : Spin n→ℝ, |
| variance g (observationField T p.1 p.2) f ≤ 2*dirichlet g (observationField T p.1 p.2) f} ≤ ENNReal.ofReal (exp (-c*(n:ℝ))) := by |
| rw [he] |
| calc |
| _ ≤ ∫⁻ _x,ENNReal.ofReal (exp (-c*(n:ℝ))) ∂μ := lintegral_mono hb |
| _ = _ := by simp |
| have hr := ENNReal.toReal_mono ENNReal.ofReal_ne_top hh' |
| simpa only [measureReal_def,ENNReal.toReal_ofReal (exp_pos _).le] using hr |
| end SKGap |
| |
| end |
| end |
| |
| end OAI |
| |
