| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGrowth |
| import OAI.Probability.CriticalPercolation.Growth.SparseEnergyRatio |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkOperator |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkProbability |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry |
| open HarmonicAnalysis |
|
|
| variable {X : Type*} (W : Geometry X) |
|
|
| lemma exists_ball_boundary [Infinite X] (r : ℕ) : ∃ x∈W.ball (r+1), x∉W.ball r := by |
| by_contra! hbad |
| have hb (n : ℕ) : W.ball n⊆W.ball r := by |
| intro x hx |
| obtain ⟨p,hp⟩ := W.mem_around.mp hx |
| have hw : ∀ {a b : X}, (p : W.graph.Walk a b) → a∈W.ball r → b∈W.ball r := by |
| intro a b p |
| induction p with |
| | nil => exact id |
| | @cons a y b h p ih => |
| intro ha |
| apply ih |
| apply hbad |
| exact W.mem_around_add ha (W.mem_around.mpr ⟨.cons h .nil,by simp⟩) |
| exact hw p (W.self_mem_around _ _) |
| have hf : (Set.univ:Set X).Finite := (W.ball r).finite_toSet.subset (by |
| intro x _; obtain ⟨n,hn⟩ := W.exists_mem_ball x; exact hb n hn) |
| exact Set.infinite_univ hf |
| |
| lemma walkFeature_symmetric (n : ℕ) (x y : X) : W.walkFeature n x y=W.walkFeature n y x := by |
| have h := W.walkPower_symmetric n (FH.delta x) (FH.delta y) |
| rw [real_inner_comm,FH.delta_inner,FH.delta_inner] at h |
| exact h |
|
|
| lemma walkEnergy_gradient (n : ℕ) : |
| (∑ y∈W.neighbors W.root, ‖W.walkFeature n y-W.walkState n‖^2)= |
| 4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) := by |
| simp only [norm_sub_sq_real,W.walkFeature_norm,Finset.sum_sub_distrib,Finset.sum_add_distrib, |
| Finset.sum_const,nsmul_eq_mul,←Finset.mul_sum,←sum_inner,W.walkFeature_average, |
| real_inner_smul_left (F:=FH X)] |
| unfold walkEnergy |
| rw [inner_sub_right,W.lazyWalk_apply,real_inner_smul_right (F:=FH X),inner_add_right, |
| real_inner_self_eq_norm_sq,W.averaging_symmetric] |
| dsimp only [walkState,degree] |
| ring |
|
|
| lemma feature_neighbor_bound (hD : 0<W.degree) (n : ℕ) {y : X} (hy : y∈W.neighbors W.root) : |
| ‖W.walkFeature n y-W.walkState n‖≤Real.sqrt (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) := by |
| have hh := (Finset.single_le_sum (f:=fun y => ‖W.walkFeature n y-W.walkState n‖^2) |
| (fun y hy => sq_nonneg _) hy).trans_eq (W.walkEnergy_gradient n) |
| have he : 0≤4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) := |
| mul_nonneg (by positivity) (W.walkEnergy_nonneg hD _) |
| nlinarith [Real.sq_sqrt he,Real.sqrt_nonneg (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)), |
| norm_nonneg (W.walkFeature n y-W.walkState n)] |
| lemma feature_edge_bound (hD : 0<W.degree) (n : ℕ) {x y : X} (h : W.graph.Adj x y) : |
| ‖W.walkFeature n y-W.walkFeature n x‖≤Real.sqrt (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) := by |
| obtain ⟨g,rfl⟩ := W.transitive W.root x |
| have he : W.walkFeature n y-W.walkFeature n (g W.root)= |
| FH.transport g.toEquiv (W.walkFeature n (g.symm y)-W.walkState n) := by |
| rw [map_sub,W.transport_walkFeature,W.transport_walkFeature,g.apply_symm_apply] |
| rw [he,LinearIsometryEquiv.norm_map] |
| apply W.feature_neighbor_bound hD |
| exact W.mem_neighbors.mpr (by simpa using g.symm.map_adj_iff.mpr h) |
| lemma feature_walk_bound (hD : 0<W.degree) (n : ℕ) {x y : X} (p : W.graph.Walk x y) : |
| ‖W.walkFeature n y-W.walkFeature n x‖≤(p.length:ℝ)*Real.sqrt (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) := by |
| induction p with |
| | nil => simp |
| | @cons x y z h p ih => |
| have he : W.walkFeature n z-W.walkFeature n x= |
| (W.walkFeature n z-W.walkFeature n y)+(W.walkFeature n y-W.walkFeature n x) := by abel |
| rw [he] |
| have ht := norm_add_le (W.walkFeature n z-W.walkFeature n y) (W.walkFeature n y-W.walkFeature n x) |
| have hb := W.feature_edge_bound hD n h |
| simp only [SimpleGraph.Walk.length_cons,Nat.cast_add,Nat.cast_one] |
| nlinarith |
| lemma feature_ball_bound (hD : 0<W.degree) (n r : ℕ) {x : X} (hx : x∈W.ball r) : |
| ‖W.walkFeature n x-W.walkState n‖≤(r:ℝ)*Real.sqrt (4*(W.degree:ℝ)*W.walkEnergy (W.walkState n)) := by |
| obtain ⟨p,hp⟩ := W.mem_around.mp hx |
| exact (W.feature_walk_bound hD n p).trans (mul_le_mul_of_nonneg_right (by exact_mod_cast hp) (Real.sqrt_nonneg _)) |
|
|
| lemma supported_energy_bound [Infinite X] (n : ℕ) : |
| W.walkState n W.root^2≤(n+1:ℝ)^2*4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) := by |
| obtain ⟨x,hx,hx'⟩ := W.exists_ball_boundary n |
| have hp := (W.walkFeature n x-W.walkState n).norm_coordinate_le W.root |
| have hb := W.feature_ball_bound W.degree_pos n (n+1) hx |
| have hv : |(W.walkFeature n x-W.walkState n) W.root|=|W.walkState n W.root| := by |
| rw [FH.sub_apply,W.walkFeature_symmetric,W.walkState_support n hx',zero_sub,abs_neg] |
| rw [hv] at hp |
| have ht := pow_le_pow_left₀ (abs_nonneg _) (hp.trans hb) 2 |
| have he : 0≤4*(W.degree:ℝ)*W.walkEnergy (W.walkState n) := |
| mul_nonneg (by positivity) (W.walkEnergy_nonneg W.degree_pos _) |
| rw [mul_pow,Real.sq_sqrt he,sq_abs] at ht |
| push_cast at ht |
| nlinarith |
| lemma walkState_energy_pos [Infinite X] (n : ℕ) : 0<W.walkEnergy (W.walkState n) := by |
| have hp := W.walkState_root_pos n |
| have hb := W.supported_energy_bound n |
| have hn := W.walkEnergy_nonneg W.degree_pos (W.walkState n) |
| by_contra! h |
| have he := le_antisymm h hn |
| rw [he,mul_zero] at hb |
| nlinarith |
| lemma walkState_energy_lower [Infinite X] (n : ℕ) : |
| 1≤((W.ball n).card:ℝ)^2*(2*(n:ℝ)+1)^2*4*(W.degree:ℝ)*W.walkEnergy (W.walkState (2*n)) := by |
| have hn := W.walkState_norm_lower W.degree_pos n |
| have hb := W.supported_energy_bound (2*n) |
| rw [W.walkState_double_at_root] at hb |
| have hsq : 1≤((W.ball n).card:ℝ)^2*(‖W.walkState n‖^2)^2 := by |
| nlinarith [mul_self_le_mul_self (by norm_num : (0:ℝ)≤1) hn] |
| have hh := mul_le_mul_of_nonneg_left hb (sq_nonneg ((W.ball n).card:ℝ)) |
| push_cast at hh |
| nlinarith |
| lemma walkState_energy_antitone (hD : 0<W.degree) : Antitone (fun n => W.walkEnergy (W.walkState n)) := by |
| apply antitone_nat_of_succ_le |
| intro n |
| have h := W.walkEnergy_dissipation hD (W.walkState n) |
| rw [←W.walkState_succ] at h |
| nlinarith [sq_nonneg ‖W.walkState n-W.lazyWalk (W.walkState n)‖] |
|
|
| lemma walk_energy_inverse_sparse [Infinite X] (d : ℕ) |
| (hg : W.HasDegree d) : |
| ∃ C : ℝ, 0≤C ∧ ∀ R, ∃ m≥R, |
| (W.walkEnergy (W.walkState m))⁻¹≤C*(m:ℝ)^(2*d+2) := by |
| obtain ⟨C,hC⟩ := hg |
| refine ⟨36*(W.degree:ℝ)*(C:ℝ)^2,by positivity,?_⟩ |
| intro R |
| obtain ⟨n,hn,hvol⟩ := hC (max R 1) |
| have hn1 : 1≤n := (le_max_right R 1).trans hn |
| have hnR : R≤n := (le_max_left R 1).trans hn |
| refine ⟨2*n,by omega,?_⟩ |
| have hV : ((W.ball n).card:ℝ)≤(C:ℝ)*(n:ℝ)^d := by exact_mod_cast hvol |
| have hn' : (1:ℝ)≤n := by exact_mod_cast hn1 |
| have hpoly : ((W.ball n).card:ℝ)^2*(2*(n:ℝ)+1)^2*4*(W.degree:ℝ)≤ |
| (36*(W.degree:ℝ)*(C:ℝ)^2)*((2*n:ℕ):ℝ)^(2*d+2) := by |
| calc |
| _ ≤ ((C:ℝ)*(n:ℝ)^d)^2*(3*(n:ℝ))^2*4*(W.degree:ℝ) := by |
| gcongr |
| linarith |
| _ = (36*(W.degree:ℝ)*(C:ℝ)^2)*(n:ℝ)^(2*d+2) := by |
| rw [pow_add,pow_mul] |
| ring |
| _ ≤ _ := by |
| gcongr |
| linarith |
| have hb := W.walkState_energy_lower n |
| have hh := mul_le_mul_of_nonneg_right hpoly (W.walkState_energy_pos (2*n)).le |
| have hf : 1≤((36*(W.degree:ℝ)*(C:ℝ)^2)*((2*n:ℕ):ℝ)^(2*d+2))* |
| W.walkEnergy (W.walkState (2*n)) := hb.trans hh |
| exact (inv_le_iff_one_le_mul₀ (W.walkState_energy_pos (2*n))).mpr hf |
|
|
| theorem exists_walk_small_relative_residual [Infinite X] (d : ℕ) |
| (hg : W.HasDegree d) (ε : ℝ) (hε : 0<ε) : |
| ∃ n, 0<W.walkEnergy (W.walkState n) ∧ |
| ‖W.walkState n-W.lazyWalk (W.walkState n)‖^2≤ε*W.walkEnergy (W.walkState n) := by |
| obtain ⟨C,hC,hpoly⟩ := W.walk_energy_inverse_sparse d hg |
| obtain ⟨n,hn⟩ := exists_small_relative_dissipation |
| (fun n => W.walkEnergy (W.walkState n)) |
| (fun n => ‖W.walkState n-W.lazyWalk (W.walkState n)‖^2) |
| W.walkState_energy_pos (W.walkState_energy_antitone W.degree_pos) |
| (fun n => by simpa only [W.walkState_succ] using W.walkEnergy_dissipation W.degree_pos (W.walkState n)) |
| (2*d+2) C hC hpoly ε hε |
| exact ⟨n,W.walkState_energy_pos n,hn⟩ |
|
|
| end CriticalPercolation.HomogeneousAnalysis.Geometry |
| end |
| end |
|
|
| end OAI |
|
|
