| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphAction |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls |
| import OAI.Probability.CriticalPercolation.Modular.RelativeModular |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def stabilizerOrbit (x y : V) : Set V := |
| MulAction.orbit (MulAction.stabilizer G.Aut x) y |
|
|
| lemma mem_stabilizerOrbit {x y z : V} : z ∈ G.stabilizerOrbit x y ↔ |
| ∃ φ : G.Aut, φ.vertex x = x ∧ φ.vertex y = z := by |
| simp only [stabilizerOrbit, MulAction.mem_orbit_iff] |
| constructor |
| · rintro ⟨φ,hφ⟩ |
| exact ⟨φ,φ.property,hφ⟩ |
| · rintro ⟨φ,hx,hy⟩ |
| exact ⟨⟨φ,hx⟩,hy⟩ |
|
|
| lemma self_mem_stabilizerOrbit (x y : V) : y ∈ G.stabilizerOrbit x y := |
| (G.mem_stabilizerOrbit).mpr ⟨1,rfl,rfl⟩ |
|
|
| lemma stabilizerOrbit_finite (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (x y : V) : (G.stabilizerOrbit x y).Finite := by |
| obtain ⟨n,hn⟩ := G.finite_subset_vertexBall hc x (Set.finite_singleton y) |
| apply (G.vertexBall_finite hl x n).subset |
| intro z hz |
| obtain ⟨φ,hx,rfl⟩ := G.mem_stabilizerOrbit.mp hz |
| rw [←hx] |
| exact (Aut.vertexBall_mem G φ x y n).mpr (hn (Set.mem_singleton y)) |
|
|
| def suborbitCard (x y : V) : ℕ := (G.stabilizerOrbit x y).ncard |
|
|
| lemma suborbitCard_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (x y : V) : 0 < G.suborbitCard x y := |
| (Set.ncard_pos (G.stabilizerOrbit_finite hc hl x y)).mpr ⟨y,G.self_mem_stabilizerOrbit x y⟩ |
|
|
| lemma suborbitCard_eq_relIndex (x y : V) : G.suborbitCard x y = |
| (MulAction.stabilizer G.Aut y).relIndex (MulAction.stabilizer G.Aut x) := by |
| have heq : (MulAction.stabilizer G.Aut y).subgroupOf (MulAction.stabilizer G.Aut x) = |
| MulAction.stabilizer (MulAction.stabilizer G.Aut x) y := by |
| ext φ |
| rfl |
| rw [Subgroup.relIndex,heq,MulAction.index_stabilizer] |
| rfl |
|
|
| lemma Aut.stabilizerOrbit_image (φ : G.Aut) (x y : V) : |
| φ.vertex '' G.stabilizerOrbit x y = G.stabilizerOrbit (φ.vertex x) (φ.vertex y) := by |
| ext z |
| constructor |
| · rintro ⟨w,hw,rfl⟩ |
| obtain ⟨ψ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hw |
| apply G.mem_stabilizerOrbit.mpr |
| refine ⟨φ * ψ * φ⁻¹,?_,?_⟩ |
| · change φ.vertex (ψ.vertex (φ.vertex.symm (φ.vertex x))) = φ.vertex x |
| rw [Equiv.symm_apply_apply,hx] |
| · change φ.vertex (ψ.vertex (φ.vertex.symm (φ.vertex y))) = φ.vertex w |
| rw [Equiv.symm_apply_apply,hy] |
| · intro hz |
| obtain ⟨ψ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hz |
| refine ⟨φ.vertex.symm z,?_,φ.vertex.apply_symm_apply z⟩ |
| apply G.mem_stabilizerOrbit.mpr |
| refine ⟨φ⁻¹ * ψ * φ,?_,?_⟩ |
| · change φ.vertex.symm (ψ.vertex (φ.vertex x)) = x |
| rw [hx,Equiv.symm_apply_apply] |
| · change φ.vertex.symm (ψ.vertex (φ.vertex y)) = φ.vertex.symm z |
| rw [hy] |
|
|
| lemma Aut.suborbitCard (φ : G.Aut) (x y : V) : |
| G.suborbitCard (φ.vertex x) (φ.vertex y) = G.suborbitCard x y := by |
| rw [CriticalPercolation.BondGraph.suborbitCard,←Aut.stabilizerOrbit_image G φ, |
| Set.ncard_image_of_injective _ φ.vertex.injective] |
| rfl |
|
|
| def modularRatio (x y : V) : ℝ := (G.suborbitCard x y : ℝ) / (G.suborbitCard y x : ℝ) |
|
|
| lemma modularRatio_eq_relativeMass (x y : V) : G.modularRatio x y = |
| Subgroup.relativeMass (MulAction.stabilizer G.Aut x) (MulAction.stabilizer G.Aut y) := by |
| simp only [modularRatio,suborbitCard_eq_relIndex,Subgroup.relativeMass] |
|
|
| lemma modularRatio_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (x y : V) : |
| 0 < G.modularRatio x y := |
| div_pos (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl x y)) |
| (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl y x)) |
|
|
| lemma modularRatio_cocycle (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (x y z : V) : G.modularRatio x y * G.modularRatio y z = G.modularRatio x z := by |
| simp only [modularRatio_eq_relativeMass] |
| apply Subgroup.relativeMass_cocycle |
| all_goals |
| rw [←suborbitCard_eq_relIndex] |
| exact ne_of_gt (G.suborbitCard_pos hc hl _ _) |
|
|
| lemma modularRatio_self (x : V) : G.modularRatio x x = 1 := by |
| rw [modularRatio_eq_relativeMass,Subgroup.relativeMass_self] |
|
|
| lemma Aut.modularRatio (φ : G.Aut) (x y : V) : |
| G.modularRatio (φ.vertex x) (φ.vertex y) = G.modularRatio x y := by |
| simp only [CriticalPercolation.BondGraph.modularRatio,Aut.suborbitCard] |
|
|
| def CombinatoriallyUnimodular : Prop := |
| ∀ (φ : G.Aut) x, G.suborbitCard x (φ.vertex x) = G.suborbitCard (φ.vertex x) x |
|
|
| lemma CombinatoriallyUnimodular.modularRatio (hu : G.CombinatoriallyUnimodular) |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (φ : G.Aut) (x : V) : |
| G.modularRatio x (φ.vertex x) = 1 := by |
| simp only [CriticalPercolation.BondGraph.modularRatio,hu φ x, |
| div_self (ne_of_gt (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl (φ.vertex x) x)) : |
| (G.suborbitCard (φ.vertex x) x : ℝ) ≠ 0)] |
|
|
| lemma CombinatoriallyUnimodular.rootWeight_invariant (hu : G.CombinatoriallyUnimodular) |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o : V) (φ : G.Aut) (x : V) : |
| G.modularRatio o (φ.vertex x) = G.modularRatio o x := by |
| rw [← G.modularRatio_cocycle hc hl o (φ.vertex o) (φ.vertex x), |
| CombinatoriallyUnimodular.modularRatio G hu hc hl φ o,one_mul,Aut.modularRatio] |
|
|
| lemma modularRatio_card_balance (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (o x y : V) : G.modularRatio o x * (G.suborbitCard x y : ℝ) = |
| G.modularRatio o y * (G.suborbitCard y x : ℝ) := by |
| have hh := G.modularRatio_cocycle hc hl o x y |
| unfold modularRatio at hh |
| rw [← mul_div_assoc] at hh |
| exact (div_eq_iff (ne_of_gt (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl y x)))).mp hh |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
