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7d9f027 | 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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | 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
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