| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphApproximateNeighborhoods |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.HomogeneousAnalysis |
| structure Geometry (X : Type*) where |
| graph : SimpleGraph X |
| connected : graph.Connected |
| locallyFinite : ∀ x, (graph.neighborSet x).Finite |
| transitive : ∀ x y, ∃ g : graph ≃g graph, g x=y |
| root : X |
|
|
| namespace Geometry |
| variable {X : Type*} (W : Geometry X) |
| abbrev Aut := W.graph ≃g W.graph |
|
|
| def around (x : X) (r : ℕ) : Finset X := (GraphTopology.reachBall_finite W.graph W.locallyFinite x r).toFinset |
| abbrev ball (r : ℕ) := W.around W.root r |
|
|
| @[simp] lemma mem_around {x y : X} {r : ℕ} : |
| y∈W.around x r ↔ y∈GraphTopology.reachBall W.graph x r := (GraphTopology.reachBall_finite W.graph W.locallyFinite x r).mem_toFinset |
|
|
| lemma around_mono (x : X) : Monotone (W.around x) := by |
| intro r s h y hy |
| exact W.mem_around.mpr (GraphTopology.reachBall_mono W.graph x h (W.mem_around.mp hy)) |
| lemma ball_mono : Monotone W.ball := W.around_mono W.root |
|
|
| lemma self_mem_around (x : X) (r : ℕ) : x∈W.around x r := |
| W.mem_around.mpr (GraphTopology.reachBall_self W.graph x r) |
| lemma mem_around_symm {x y : X} {r : ℕ} (h : y∈W.around x r) : x∈W.around y r := |
| W.mem_around.mpr (GraphTopology.reachBall_symm W.graph (W.mem_around.mp h)) |
| lemma mem_around_add {x y z : X} {r s : ℕ} (hy : y∈W.around x r) (hz : z∈W.around y s) : |
| z∈W.around x (r+s) := |
| W.mem_around.mpr (GraphTopology.reachBall_add W.graph (W.mem_around.mp hy) (W.mem_around.mp hz)) |
| lemma around_subset_ball {x : X} {R r : ℕ} (hx : x∈W.ball R) : W.around x r⊆W.ball (R+r) := |
| fun _ hy => W.mem_around_add hx hy |
| lemma ball_card_pos (r : ℕ) : 0<(W.ball r).card := |
| Finset.card_pos.mpr ⟨W.root,W.self_mem_around W.root r⟩ |
|
|
| lemma aut_around_image (g : W.Aut) (x : X) (r : ℕ) : |
| (W.around x r).image g=W.around (g x) r := by |
| ext y |
| constructor |
| · intro hy |
| obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hy |
| exact W.mem_around.mpr (GraphTopology.iso_reachBall W.graph g (W.mem_around.mp hz)) |
| · intro hy |
| refine Finset.mem_image.mpr ⟨g.symm y,?_,g.apply_symm_apply y⟩ |
| apply W.mem_around.mpr |
| simpa using GraphTopology.iso_reachBall W.graph g.symm (W.mem_around.mp hy) |
|
|
| lemma card_around (x : X) (r : ℕ) : (W.around x r).card=(W.ball r).card := by |
| obtain ⟨g,rfl⟩ := W.transitive W.root x |
| rw [←W.aut_around_image] |
| exact Finset.card_image_of_injective _ g.injective |
|
|
| lemma exists_mem_ball (x : X) : ∃ n, x∈W.ball n := by |
| obtain ⟨p,hp,he⟩ := W.connected.exists_path_of_dist W.root x |
| exact ⟨p.length,W.mem_around.mpr ⟨p,le_rfl⟩⟩ |
|
|
| def length (x : X) : ℕ := W.graph.dist W.root x |
| lemma mem_ball_length (x : X) : x∈W.ball (W.length x) := by |
| obtain ⟨p,_,hp⟩ := W.connected.exists_path_of_dist W.root x |
| exact W.mem_around.mpr ⟨p,hp.le⟩ |
| lemma length_le_iff {x : X} {n : ℕ} : W.length x≤n ↔ x∈W.ball n := by |
| constructor |
| · intro h; exact W.ball_mono h (W.mem_ball_length x) |
| · intro h |
| obtain ⟨p,hp⟩ := W.mem_around.mp h |
| exact (SimpleGraph.dist_le p).trans hp |
| lemma length_root : W.length W.root=0 := SimpleGraph.dist_self |
| lemma length_adj {x y : X} (h : W.graph.Adj x y) : |
| W.length x≤W.length y+1 ∧ W.length y≤W.length x+1 := by |
| have h1 := W.connected.dist_triangle (u:=W.root) (v:=x) (w:=y) |
| have h2 := W.connected.dist_triangle (u:=W.root) (v:=y) (w:=x) |
| rw [SimpleGraph.dist_eq_one_iff_adj.mpr h] at h1 |
| rw [SimpleGraph.dist_eq_one_iff_adj.mpr h.symm] at h2 |
| exact ⟨h2,h1⟩ |
|
|
| lemma aut_length_le (g : W.Aut) (x : X) : W.length (g x)≤W.length (g W.root)+W.length x := by |
| apply W.length_le_iff.mpr |
| apply W.mem_around_add (W.mem_ball_length (g W.root)) |
| rw [←W.aut_around_image] |
| exact Finset.mem_image.mpr ⟨x,W.mem_ball_length x,rfl⟩ |
|
|
| def neighbors (x : X) : Finset X := (W.locallyFinite x).toFinset |
| @[simp] lemma mem_neighbors {x y : X} : y∈W.neighbors x ↔ W.graph.Adj x y := (W.locallyFinite x).mem_toFinset |
| lemma neighbors_image (g : W.Aut) (x : X) : (W.neighbors x).image g=W.neighbors (g x) := by |
| ext y |
| simp only [Finset.mem_image,W.mem_neighbors] |
| constructor |
| · rintro ⟨z,hz,rfl⟩; exact g.map_adj_iff.mpr hz |
| · intro h; exact ⟨g.symm y,by simpa using g.symm.map_adj_iff.mpr h,g.apply_symm_apply y⟩ |
| lemma neighbors_card (x : X) : (W.neighbors x).card=(W.neighbors W.root).card := by |
| obtain ⟨g,rfl⟩ := W.transitive W.root x |
| rw [←W.neighbors_image] |
| exact Finset.card_image_of_injective _ g.injective |
| abbrev degree := (W.neighbors W.root).card |
|
|
| def energy (F : Finset X) (u : X → ℝ) := ∑ x∈F, ∑ y∈W.neighbors x, (u y-u x)^2 |
| lemma energy_nonneg (F : Finset X) (u : X → ℝ) : 0≤W.energy F u := by |
| exact Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
| lemma energy_mono {F H : Finset X} (h : F⊆H) (u : X → ℝ) : W.energy F u≤W.energy H u := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg h |
| intros; exact Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| |
| def IsHarmonic (u : X → ℝ) : Prop := ∀ x, (∑ y∈W.neighbors x, (u x-u y))=0 |
|
|
| def cutoff (R : ℕ) (x : X) : ℝ := max 0 (2*(R:ℝ)-(W.length x:ℝ)) |
| lemma cutoff_nonneg (R : ℕ) (x : X) : 0≤W.cutoff R x := le_max_left _ _ |
| lemma cutoff_large (R : ℕ) {x : X} (hx : x∈W.ball R) : (R:ℝ)≤W.cutoff R x := by |
| have hl : (W.length x:ℝ)≤R := by exact_mod_cast W.length_le_iff.mpr hx |
| exact (by linarith : (R:ℝ)≤2*R-W.length x).trans (le_max_right _ _) |
| lemma cutoff_eq_zero (R : ℕ) {x : X} (hx : 2*R≤W.length x) : W.cutoff R x=0 := by |
| have hl : 2*(R:ℝ)≤W.length x := by exact_mod_cast hx |
| exact max_eq_left (by linarith) |
| lemma cutoff_ne_zero_length (R : ℕ) {x : X} (hx : W.cutoff R x≠0) : W.length x<2*R := by |
| by_contra! h; exact hx (W.cutoff_eq_zero R h) |
| lemma cutoff_lipschitz (R : ℕ) {x y : X} (h : W.graph.Adj x y) : |
| |W.cutoff R y-W.cutoff R x|≤1 := by |
| obtain ⟨h1,h2⟩ := W.length_adj h |
| have h1' : (W.length x:ℝ)≤(W.length y:ℝ)+1 := by exact_mod_cast h1 |
| have h2' : (W.length y:ℝ)≤(W.length x:ℝ)+1 := by exact_mod_cast h2 |
| unfold cutoff |
| rw [abs_le] |
| constructor <;> rcases le_total 0 (2*(R:ℝ)-W.length y) with hx|hx <;> |
| rcases le_total 0 (2*(R:ℝ)-W.length x) with hy|hy <;> |
| simp only [max_eq_right hx,max_eq_left hx,max_eq_right hy,max_eq_left hy] <;> linarith |
| |
| end Geometry |
| end CriticalPercolation.HomogeneousAnalysis |
| end |
| end |
| |
| end OAI |
| |
