| import Mathlib |
| import OAI.Probability.CriticalPercolation.Clusters.HitValue |
| import OAI.Probability.CriticalPercolation.Gluing.FiniteJointGluingStrict |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ContinuousExpectWeights |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped BigOperators |
| open Set MeasureTheory ProbabilityTheory |
|
|
| namespace CriticalPercolation.BondGraph |
| open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion |
| open CriticalPercolation.ClusterColumns |
| open scoped Classical |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E) |
|
|
| lemma finite_joint_gluing (w : E → Prop → ℝ) |
| (hw : ∀ e, ∑ b, w e b = 1) (hpos : ∀ e b, 0 ≤ w e b) |
| (o : V) (A T : Set V) (a : ℝ) |
| (ha : ∀ v ∈ A, a ≤ expect w |
| (fun ω => hitValue T (component (G.openGraph ω) v))) : |
| expect w (fun ω => hitValue A (component (G.openGraph ω) o)) * a ≤ |
| expect w (fun ω => jointHitValue A T (component (G.openGraph ω) o)) := by |
| classical |
| let w' : ℝ → E → Prop → ℝ := fun t e b => (1-t) * w e b + if b then 0 else t |
| have hzero : w' 0 = w := by funext e b; simp [w'] |
| have hcont : Continuous w' := by |
| apply continuous_pi |
| intro e |
| apply continuous_pi |
| intro b |
| dsimp only [w'] |
| split_ifs <;> fun_prop |
| let err : ℝ → ℝ := fun t => ∑ v : V, |
| |expect (w' t) (fun ω => hitValue T (component (G.openGraph ω) v)) - |
| expect w (fun ω => hitValue T (component (G.openGraph ω) v))| |
| have herr : Continuous err := by |
| apply continuous_finsetSum |
| intro v _ |
| exact (((continuous_expect_weights _).comp hcont).sub continuous_const).abs |
| have herr0 : err 0 = 0 := by simp [err, hzero] |
| have hbound (t : ℝ) (ht : 0 < t) (ht1 : t ≤ 1) : |
| expect (w' t) (fun ω => hitValue A (component (G.openGraph ω) o)) * (a-err t) ≤ |
| expect (w' t) (fun ω => jointHitValue A T (component (G.openGraph ω) o)) := by |
| apply G.finite_joint_gluing_strict |
| · intro e |
| have hs := hw e |
| simp only [Fintype.sum_Prop] at hs ⊢ |
| dsimp only [w'] |
| rw [ite_eq_left True.intro, ite_eq_right (by trivial : ¬ False)] |
| nlinarith |
| · intro e b |
| have hm := mul_nonneg (sub_nonneg.mpr ht1) (hpos e b) |
| dsimp only [w'] |
| apply add_nonneg hm |
| split_ifs <;> linarith |
| · intro e |
| have hm := mul_nonneg (sub_nonneg.mpr ht1) (hpos e False) |
| dsimp only [w'] |
| rw [ite_eq_right (by trivial : ¬ False)] |
| exact add_pos_of_nonneg_of_pos hm ht |
| · intro v hv |
| have hsingle : |
| |expect (w' t) (fun ω => hitValue T (component (G.openGraph ω) v)) - |
| expect w (fun ω => hitValue T (component (G.openGraph ω) v))| ≤ err t := by |
| dsimp only [err] |
| exact Finset.single_le_sum (f := fun u : V => |
| |expect (w' t) (fun ω => hitValue T (component (G.openGraph ω) u)) - |
| expect w (fun ω => hitValue T (component (G.openGraph ω) u))|) |
| (fun _ _ => abs_nonneg _) (Finset.mem_univ v) |
| have hneg := (abs_le.mp hsingle).1 |
| linarith [ha v hv] |
| have ht : Filter.Tendsto (fun n : ℕ => 1 / ((n : ℝ)+1)) Filter.atTop (nhds 0) := |
| tendsto_one_div_add_atTop_nhds_zero_nat |
| have hL := ((((continuous_expect_weights |
| (fun ω => hitValue A (component (G.openGraph ω) o))).comp hcont).mul |
| ((continuous_const (y := a)).sub herr)).tendsto 0).comp ht |
| have hR := (((continuous_expect_weights |
| (fun ω => jointHitValue A T (component (G.openGraph ω) o))).comp hcont).tendsto 0).comp ht |
| have hh := le_of_tendsto_of_tendsto hL hR (Filter.Eventually.of_forall (fun n : ℕ => |
| hbound (1/((n : ℝ)+1)) (by positivity) (by |
| apply (div_le_one (by positivity : (0 : ℝ) < n+1)).mpr |
| have hn := Nat.cast_nonneg (α := ℝ) n |
| linarith))) |
| simpa only [Pi.mul_apply, Pi.sub_apply, Function.comp_apply, hzero, herr0, sub_zero] using hh |
|
|
| end CriticalPercolation.BondGraph |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| end |
| end |
|
|
| end OAI |
|
|
