| import Mathlib |
| import OAI.Probability.CriticalPercolation.Gluing.ContourSites |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.Boundary |
|
|
| def wordDir {n : ℕ} (w : Fin n → Dir) (i : ℕ) : Dir := |
| if h : i<n then w ⟨i,h⟩ else if h₀ : 0<n then w ⟨0,h₀⟩ else 0 |
|
|
| def wordEdge {n : ℕ} (p : Grid) (w : Fin n → Dir) (i : ℕ) : Edge := |
| (p+∑ j∈Finset.range i, step (wordDir w j),wordDir w i) |
|
|
| lemma wordEdge_chain {n : ℕ} (p : Grid) (w : Fin n → Dir) (i : ℕ) : |
| head (wordEdge p w i)=tail (wordEdge p w (i+1)) := by |
| simp [wordEdge,head,tail,Finset.sum_range_succ,add_assoc] |
|
|
| lemma wordEdge_zero {n : ℕ} (p : Grid) (w : Fin n → Dir) : |
| (wordEdge p w 0).1=p := by simp [wordEdge] |
|
|
| lemma wordEdge_reconstruct {n : ℕ} (hn : 0<n) (e : ℕ → Edge) |
| (hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0) : |
| ∀ i≤n, wordEdge (e 0).1 (fun j : Fin n => (e j).2) i=e i := by |
| let w : Fin n → Dir := fun j => (e j).2 |
| have hd (i : ℕ) (hi : i<n) : wordDir w i=(e i).2 := by simp [wordDir,hi,w] |
| have hv : ∀ i≤n, (wordEdge (e 0).1 w i).1=(e i).1 := by |
| intro i hi |
| induction i with |
| | zero => exact wordEdge_zero _ _ |
| | succ i ih => |
| have hh : i<n := by omega |
| have ih' := ih (by omega) |
| have hc := wordEdge_chain (e 0).1 w i |
| change (wordEdge (e 0).1 w i).1+step (wordDir w i)= |
| (wordEdge (e 0).1 w (i+1)).1 at hc |
| rw [← hc,ih',hd i hh] |
| exact hchain i |
| intro i hi |
| apply Prod.ext (hv i hi) |
| change wordDir w i=(e i).2 |
| by_cases hlt : i<n |
| · exact hd i hlt |
| · have hie : i=n := by omega |
| subst i |
| simp [wordDir,hn,hclose,w] |
|
|
| lemma card_words (n : ℕ) : Fintype.card (Fin n → Dir)=4^n := by |
| simp [Dir] |
|
|
| def wordValid {n : ℕ} (p : Grid) (w : Fin n → Dir) : Prop := |
| wordEdge p w n=wordEdge p w 0 ∧ Set.InjOn (wordEdge p w) (Set.Iio n) |
|
|
| lemma wordValid_reconstruct {n : ℕ} (hn : 0<n) (e : ℕ → Edge) |
| (hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0) |
| (hinj : Set.InjOn e (Set.Iio n)) : |
| wordValid (e 0).1 (fun j : Fin n => (e j).2) := by |
| have hr := wordEdge_reconstruct hn e hchain hclose |
| refine ⟨?_,?_⟩ |
| · rw [hr n le_rfl,hr 0 (Nat.zero_le n),hclose] |
| · intro i hi j hj he |
| rw [hr i hi.le,hr j hj.le] at he |
| exact hinj hi hj he |
|
|
| lemma outsideSites_congr {n : ℕ} {e f : ℕ → Edge} |
| (h : ∀ i<n, e i=f i) : outsideSites e n=outsideSites f n := by |
| have hf : forwardIndices e n=forwardIndices f n := by |
| apply Finset.filter_congr |
| intro i hi |
| rw [h i (Finset.mem_range.mp hi)] |
| unfold outsideSites |
| rw [hf] |
| apply Finset.image_congr |
| intro i hi |
| dsimp only |
| rw [h i (Finset.mem_range.mp (Finset.mem_filter.mp hi).1)] |
|
|
| theorem finite_good_contour (S : Set Grid) (hS : S.Finite) (hroot : (0,0)∈S) |
| (hquad : ∀ v∈S, 0≤v.1 ∧ 0≤v.2) : |
| ∃ n : ℕ, ∃ w : Fin (n+1) → Dir, |
| wordValid (0,1) w ∧ |
| (∀ z∈outsideSites (wordEdge (0,1) w) (n+1), |
| z∉S ∧ ∃ y∈S, z=y+(1,0) ∨ z=y+(0,1)) := by |
| obtain ⟨l,hl,e,h0,hend,hmem,hchain,hinj⟩ := squareBoundary_closed_trail hS |
| ((0,1),3) (west_root_edge_mem S hroot hquad) |
| obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hl) |
| have he : e (n+1)=e 0 := hend.trans h0.symm |
| have hp : (e 0).1=(0,1) := congrArg Prod.fst h0 |
| refine ⟨n,(fun j : Fin (n+1) => (e j).2),?_,?_⟩ |
| · simpa only [hp] using wordValid_reconstruct hl e hchain he hinj |
| · have hr := wordEdge_reconstruct hl e hchain he |
| rw [hp] at hr |
| rw [outsideSites_congr (fun i hi => hr i hi.le)] |
| exact outsideSites_failed_children S e (n+1) hmem |
|
|
| end CriticalPercolation.Boundary |
| end |
| end |
| end |
|
|
| end OAI |
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|
