File size: 3,582 Bytes
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 | import Mathlib
import OAI.Probability.SKGap.Localization.FixedTimeQuenched
import OAI.Probability.SKGap.Localization.OperatorQuenched
import OAI.Probability.SKGap.Localization.GridSteps
import OAI.Probability.SKGap.Localization.QuenchedHelpers
namespace OAI
namespace SKGap.FullMain
open MeasureTheory ProbabilityTheory Filter Real Set ObservationBridge GaussianHistory
open scoped Topology BigOperators ENNReal
noncomputable section
def gridFailure (n : ℕ) (j A K ε c ρ T : ℝ) (g : Disorder n) : ℝ :=
∑k∈Finset.range (gridSteps n+1),fixedTimeFailure n j A K ε c ρ ((k:ℝ)*gridStep T n) g
lemma gridFailure_nonneg (n : ℕ) (j A K ε c ρ T : ℝ) (g : Disorder n) :
0≤gridFailure n j A K ε c ρ T g :=
Finset.sum_nonneg (fun step _=>fixedTimeFailure_nonneg n j A K ε c ρ ((step:ℝ)*gridStep T n) g)
lemma weighted_gridFailure_integrable (β : ℝ) (n : ℕ) (A K ε c ρ T : ℝ) :
Integrable (fun g=>partitionSizeBias β n g*gridFailure n (β^2) A K ε c ρ T g) (disorderLaw β n) := by
simp only [gridFailure,Finset.mul_sum]
exact integrable_finsetSum _ (fun k _=>weighted_fixedTimeFailure_integrable _ _ _ _ _ _ _ _ _)
lemma grid_time_mem {T : ℝ} (hT : 0≤T) {n k : ℕ} (hk : k≤gridSteps n) :
(k:ℝ)*gridStep T n∈Icc 0 T := by
have ht:=gridStep_nonneg hT n
refine ⟨mul_nonneg (Nat.cast_nonneg _) ht,?_⟩
exact (mul_le_mul_of_nonneg_right (by exact_mod_cast hk) ht).trans_eq (grid_total T n)
theorem quenched_grid_root_stability {β T : ℝ} (hβ : 0<β) (hβ1 : β<1) (hT : 0<T) :
∃A K ε c ρ a : ℝ,1<A ∧ 2*β<K ∧ β*A<1 ∧ 0<ε ∧ 0<c ∧ 0<ρ ∧ 0<a ∧
Whp β (fun n g=>operatorBound K (coupling g) ∧
gridFailure n (β^2) A K ε c ρ T g≤exp (-a*(n:ℝ))) := by
obtain ⟨A,K,ε,c,ρ,a,hA,hK,hs,hε,hc,hρ,ha,N,hN,hann⟩:=
annealed_fixed_time_root_stability hβ hβ1 hT
have hsum : ∀ᶠ n:ℕ in atTop,
(∫g,partitionSizeBias β n g*gridFailure n (β^2) A K ε c ρ T g ∂disorderLaw β n)≤
exp (-(a/2)*(n:ℝ)) := by
filter_upwards [eventually_ge_atTop N,grid_exp_absorb ha] with n hn he
simp only [gridFailure,Finset.mul_sum]
rw [integral_finsetSum _ (fun k _=>weighted_fixedTimeFailure_integrable _ _ _ _ _ _ _ _ _)]
have he' : (∑_k∈Finset.range (gridSteps n+1),exp (-a*(n:ℝ)))≤exp (-(a/2)*(n:ℝ)) := by
simpa only [Finset.sum_const,Finset.card_range,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] using he
apply le_trans (Finset.sum_le_sum (fun k hk=>?_)) he'
have hk' : k≤gridSteps n:=by have:=Finset.mem_range.mp hk;omega
exact weighted_fixedTimeFailure_integral_bound β n A K ε c ρ ((k:ℝ)*gridStep T n) _
(exp_pos _).le (hann n hn _ (grid_time_mem hT.le hk'))
have hp : Whp β (fun n g=>gridFailure n (β^2) A K ε c ρ T g≤exp (-((a/2)/4)*(n:ℝ))) := by
apply (whp_iff_real _ _).mpr
simp only [not_le]
apply exponential_size_bias_transfer (C:=1) (disorderLaw β) (partitionSizeBias β)
(fun n=>gridFailure n (β^2) A K ε c ρ T)
(fun n g=>(partitionSizeBias_pos β n g).le)
(fun n g=>gridFailure_nonneg _ _ _ _ _ _ _ _ _) (fun n=>weighted_gridFailure_integrable _ _ _ _ _ _ _ _)
(half_pos ha) zero_le_one
· exact disorder_partition_lower_tail hβ hβ1 (by positivity)
· simpa only [one_mul] using hsum
have hop : Whp β (fun n g=>operatorBound K (coupling g)):=
(whp_iff_real _ _).mpr (quenched_operator_bound hβ hK)
exact ⟨A,K,ε,c,ρ,(a/2)/4,hA,hK,hs,hε,hc,hρ,by positivity,hop.and hp⟩
end
end SKGap.FullMain
end OAI
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