| 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 |
|
|
