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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.EnergyTail
import OAI.Probability.SKBarriers.Interpolation.RestrictedConcentration
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem integral_le_off_exception {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ] {f : Ω → ℝ}
(hf : Integrable f μ) {A : Set Ω} (hA : MeasurableSet A)
{r : ℝ} (hr : 0 ≤ r) (h1 : ∀ x, f x ≤ 1)
(hout : ∀ x ∉ A, f x ≤ r) : ∫ x, f x ∂μ ≤ r+μ.real A := by
calc
_ ≤ ∫ x, r+A.indicator (fun _ => (1:ℝ)) x ∂μ := by
apply integral_mono hf ((integrable_const r).add ((integrable_const (1:ℝ)).indicator hA))
intro x
by_cases hx : x ∈ A
· simp only [Pi.add_apply,Set.indicator_of_mem hx]
linarith [h1 x]
· simp only [Pi.add_apply,Set.indicator_of_notMem hx,add_zero]
exact hout x hx
_ = _ := by
rw [integral_add (integrable_const r) ((integrable_const (1:ℝ)).indicator hA),
integral_const,integral_indicator_const _ hA]
simp only [probReal_univ,smul_eq_mul,mul_one,one_mul]
theorem restrictedLogPartition_upper_tail_linear {n d : ℕ} (hn : 0 < n) (hd : 0 < d)
{β a : ℝ} (hβ : β ≠ 0) (ha : 0 ≤ a)
{S : Finset (ReplicaConfig n d)} (hS : S.Nonempty) :
(disorderLaw n).real {J | a*(n:ℝ) ≤ restrictedLogPartition β S J-
∫ K, restrictedLogPartition β S K ∂disorderLaw n} ≤
Real.exp (-(4*a^2/(Real.pi^2*β^2*(d:ℝ)^2))*(n:ℝ)) := by
have H := restrictedLogPartition_upper_tail hn hd β (a*(n:ℝ)) hS (by positivity)
convert H using 1
congr 1
change -(4*a^2/(Real.pi^2*β^2*(d:ℝ)^2))*(n:ℝ) =
-(a*(n:ℝ))^2/(2*(Real.pi^2*(β*(d:ℝ))^2*(n:ℝ)/8))
have hn' : (n:ℝ) ≠ 0 := by exact_mod_cast hn.ne'
have hd' : (d:ℝ) ≠ 0 := by exact_mod_cast hd.ne'
field_simp
ring
theorem replicaGibbsMass_integral_le_of_mean_gap {n d : ℕ} (hn : 0 < n) (hd : 0 < d)
{β η : ℝ} (hβ : β ≠ 0) (hη : 0 ≤ η)
{S : Finset (ReplicaConfig n d)} (hS : S.Nonempty)
(hgap : (∫ J, restrictedLogPartition β S J ∂disorderLaw n)-
(d:ℝ)*(∫ J, logPartition β J ∂disorderLaw n) ≤ -(n:ℝ)*η/2) :
(∫ J, replicaGibbsMass β J S ∂disorderLaw n) ≤
Real.exp (-(n:ℝ)*η/4)+2*Real.exp (-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2)) := by
let X := restrictedLogPartition β S
let EX := ∫ J, X J ∂disorderLaw n
let EY := ∫ J, logPartition β J ∂disorderLaw n
let A : Set (Disorder n) := {J | (η/8)*(n:ℝ) ≤ X J-EX}
let B : Set (Disorder n) := {J | (η/(8*(d:ℝ)))*(n:ℝ) ≤ EY-logPartition β J}
have hA : MeasurableSet A := isClosed_le continuous_const
((continuous_restrictedLogPartition β hS).sub continuous_const) |>.measurableSet
have hB : MeasurableSet B := isClosed_le continuous_const
(continuous_const.sub (continuous_logPartition β)) |>.measurableSet
have hd0 : 0 < (d:ℝ) := by exact_mod_cast hd
have hdn : (d:ℝ) ≠ 0 := ne_of_gt hd0
have HA := restrictedLogPartition_upper_tail_linear hn hd hβ (show 0 ≤ η/8 by positivity) hS
have HB := logPartition_lower_tail_linear hn hβ (show 0 ≤ η/(8*(d:ℝ)) by positivity)
have heA : -(4*(η/8)^2/(Real.pi^2*β^2*(d:ℝ)^2))*(n:ℝ) =
-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2) := by ring
have heB : -(4*(η/(8*(d:ℝ)))^2/(Real.pi^2*β^2))*(n:ℝ) =
-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2) := by field_simp; ring
rw [heA] at HA
rw [heB] at HB
have H := integral_le_off_exception (replicaGibbsMass_integrable β S) (hA.union hB)
(Real.exp_pos (-(n:ℝ)*η/4)).le (fun J => replicaGibbsMass_le_one β J S) ?_
· have hAB : (disorderLaw n).real (A∪B) ≤
2*Real.exp (-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2)) := by
apply (measureReal_union_le (μ := disorderLaw n) A B).trans
change (disorderLaw n).real A ≤ _ at HA
change (disorderLaw n).real B ≤ _ at HB
linarith
linarith
· intro J hJ
rw [Set.mem_union,not_or] at hJ
have ha : X J-EX < (η/8)*(n:ℝ) := lt_of_not_ge hJ.1
have hb : EY-logPartition β J < (η/(8*(d:ℝ)))*(n:ℝ) := lt_of_not_ge hJ.2
have hb' := mul_lt_mul_of_pos_left hb hd0
have he : (d:ℝ)*((η/(8*(d:ℝ)))*(n:ℝ)) = η*(n:ℝ)/8 := by field_simp
rw [he] at hb'
rw [replicaGibbsMass_eq_exp β J hS]
apply Real.exp_le_exp.mpr
change EX-(d:ℝ)*EY ≤ -(n:ℝ)*η/2 at hgap
change restrictedLogPartition β S J-EX < _ at ha
linarith
end SK.Analytic
end
end
end
end OAI
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