| import Mathlib |
| import OAI.Probability.CriticalPercolation.Modular.ModularClusterMass |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical ENNReal BigOperators |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def bondWeight (o : V) (e : E) : ℝ≥0∞ := ∑ v ∈ (G.ends e).toFinset, G.stabilizerWeight o v |
|
|
| lemma bondWeight_ne_top (o : V) (e : E) : G.bondWeight o e ≠ ∞ := by |
| apply ENNReal.sum_ne_top.mpr |
| intro v hv |
| exact G.stabilizerWeight_ne_top o v |
|
|
| lemma bondWeight_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (o a : V) (e : E) : G.bondWeight o e = G.stabilizerWeight o a * G.bondWeight a e := by |
| unfold bondWeight |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro v hv |
| exact G.stabilizerWeight_change_base hc hl o a v |
|
|
| lemma Aut.bondWeight (φ : G.Aut) (o : V) (e : E) : |
| G.bondWeight (φ.vertex o) (φ.edge e) = G.bondWeight o e := by |
| unfold CriticalPercolation.BondGraph.bondWeight |
| rw [φ.preserves] |
| have himg : (Sym2.map φ.vertex (G.ends e)).toFinset = (G.ends e).toFinset.image φ.vertex := by |
| ext v |
| simp only [Sym2.mem_toFinset,Sym2.mem_map,Finset.mem_image] |
| rw [himg,Finset.sum_image] |
| · simp only [Aut.stabilizerWeight] |
| · exact fun _ _ _ _ h => φ.vertex.injective h |
|
|
| lemma incidence_stabilizerOrbit_subset_ball (v w : V) (n : ℕ) (hw : w∈G.vertexBall v n) : |
| G.stabilizerOrbit v w ⊆ G.vertexBall v n := by |
| intro z hz |
| obtain ⟨φ,hv,hwz⟩ := G.mem_stabilizerOrbit.mp hz |
| have h := (Aut.vertexBall_mem G φ v w n).mpr hw |
| simpa only [hv,hwz] using h |
|
|
| lemma stabilizerWeight_le_ball_bound (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (K : ℕ) (hK : ∀ v, (G.vertexBall v 1).ncard≤K) {v w : V} |
| (_hw : w∈G.vertexBall v 1) (hv : v∈G.vertexBall w 1) : |
| G.stabilizerWeight v w ≤ K := by |
| unfold stabilizerWeight modularRatio |
| apply ENNReal.ofReal_le_of_le_toReal |
| simp only [ENNReal.toReal_natCast] |
| apply (div_le_iff₀ (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl v w))).mpr |
| have hnum : G.suborbitCard w v ≤ K := |
| (Set.ncard_le_ncard (G.incidence_stabilizerOrbit_subset_ball w v 1 hv) |
| (G.vertexBall_finite hl w 1)).trans (hK w) |
| have hden : (1:ℝ)≤G.suborbitCard v w := by exact_mod_cast G.suborbitCard_pos hc hl v w |
| exact (Nat.cast_le.mpr hnum).trans (by nlinarith [show (0:ℝ)≤K by positivity]) |
|
|
| lemma common_bond_ball_one {v w : V} {e : E} (hv : v∈G.ends e) (hw : w∈G.ends e) : |
| w∈G.vertexBall v 1 := by |
| by_cases he : v=w |
| · subst w |
| exact G.mem_vertexBall_self v 1 |
| · apply G.vertexBall_step (G.mem_vertexBall_self v 0) |
| apply (SimpleGraph.fromEdgeSet_adj _).mpr |
| refine ⟨?_,he⟩ |
| refine ⟨e,Set.mem_univ e,?_⟩ |
| exact (Sym2.mem_and_mem_iff he).mp ⟨hv,hw⟩ |
|
|
| lemma bondWeight_le_at_end (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (K : ℕ) (hK : ∀ v, (G.vertexBall v 1).ncard≤K) (o v : V) (e : E) |
| (hv : v∈G.ends e) : G.bondWeight o e ≤ (2*K : ℕ)*G.stabilizerWeight o v := by |
| rw [G.bondWeight_change_base hc hl o v] |
| have hsum : G.bondWeight v e ≤ (2*K : ℕ) := by |
| calc |
| _ ≤ ∑ w∈(G.ends e).toFinset, (K : ℝ≥0∞) := by |
| apply Finset.sum_le_sum |
| intro w hw |
| have hw' : w∈G.ends e := Sym2.mem_toFinset.mp hw |
| exact G.stabilizerWeight_le_ball_bound hc hl K hK |
| (G.common_bond_ball_one hv hw') (G.common_bond_ball_one hw' hv) |
| _ = ((G.ends e).toFinset.card * K : ℕ) := by simp [nsmul_eq_mul] |
| _ ≤ (2*K : ℕ) := by |
| have hh : (G.ends e).toFinset.card≤2 := by |
| rw [Sym2.card_toFinset] |
| split_ifs <;> omega |
| exact_mod_cast Nat.mul_le_mul_right K hh |
| calc _ ≤ G.stabilizerWeight o v * (2*K : ℕ) := by gcongr |
| _ = _ := mul_comm _ _ |
|
|
| lemma weighted_incidence_le (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) |
| (hK : ∀ v, (G.vertexBall v 1).ncard≤K) (o : V) (S : Set V) : |
| (∑' e : E, if e∈G.incidentEdges S then G.bondWeight o e else 0) ≤ |
| (D*(2*K) : ℕ)*G.modularMass o S := by |
| calc |
| _ ≤ ∑' e : E, ∑' v : V, if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0 := by |
| apply ENNReal.tsum_le_tsum |
| intro e |
| by_cases he : e∈G.incidentEdges S |
| · rw [ite_eq_left he] |
| obtain ⟨v,hv,hve⟩ := he |
| simpa only [ite_eq_left (show v∈S ∧ v∈G.ends e from ⟨hv,hve⟩)] using |
| (ENNReal.le_tsum v (f := fun v => if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0)) |
| · simp only [ite_eq_right he]; exact zero_le |
| _ = ∑' v : V, ∑' e : E, if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0 := |
| ENNReal.tsum_comm |
| _ ≤ ∑' v : V, if v∈S then (D*(2*K) : ℕ)*G.stabilizerWeight o v else 0 := by |
| apply ENNReal.tsum_le_tsum |
| intro v |
| by_cases hv : v∈S |
| · simp only [hv,true_and,ite_true] |
| have hf : (G.incidentEdges {v}).Finite := by simpa [incidentEdges] using hl v |
| have hm (e : E) : e∈G.incidentEdges {v} ↔ v∈G.ends e := by |
| simp [incidentEdges] |
| have ht : (∑' e : E, if v∈G.ends e then G.bondWeight o e else 0) = |
| ∑ e∈hf.toFinset, G.bondWeight o e := by |
| rw [tsum_eq_sum (s := hf.toFinset) (fun e he => by |
| simp only [hf.mem_toFinset,hm] at he |
| exact ite_eq_right he)] |
| apply Finset.sum_congr rfl |
| intro e he |
| exact ite_eq_left ((hm e).mp (hf.mem_toFinset.mp he)) |
| rw [ht] |
| calc |
| _ ≤ ∑ e∈hf.toFinset, (2*K : ℕ)*G.stabilizerWeight o v := by |
| apply Finset.sum_le_sum |
| intro e he |
| exact G.bondWeight_le_at_end hc hl K hK o v e ((hm e).mp (hf.mem_toFinset.mp he)) |
| _ = ((G.incidentEdges {v}).ncard : ℝ≥0∞)*((2*K : ℕ)*G.stabilizerWeight o v) := by |
| simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf] |
| _ ≤ (D : ℝ≥0∞)*((2*K : ℕ)*G.stabilizerWeight o v) := by |
| gcongr |
| exact_mod_cast hD v |
| _ = _ := by simp only [Nat.cast_mul,mul_assoc] |
| · simp only [hv,false_and,ite_false,tsum_zero,le_refl] |
| _ = _ := by |
| unfold modularMass |
| rw [←ENNReal.tsum_mul_left] |
| apply tsum_congr |
| intro v |
| split_ifs <;> simp |
| |
| end CriticalPercolation.BondGraph |
| end |
| end |
| |
| end OAI |
| |
