openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /InfiniteAdjacent.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Exploration.BallIncidence | |
| import OAI.Probability.CriticalPercolation.Criticality.SubexponentialVolumes | |
| import OAI.Probability.CriticalPercolation.Criticality.WitnessEvents | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.UniformGeometry | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteAdjacentMeasure | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| lemma adjacent_bound_of_geometry (hconnected : G.fullGraph.Connected) | |
| (hlocal : G.LocallyFinite) (hg : G.SubexponentialGrowth) | |
| (C M D : ℕ) (hM : ∀ v, (G.vertexBall v C).ncard≤M) | |
| (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) | |
| (hdense : ∀ u v, ∃ φ : G.Aut, φ.vertex u ∈ G.vertexBall v C) | |
| (p : unitInterval) (hp : (p:ℝ)∈Set.Ioo 0 1) (e : E) (K : ℕ) : | |
| (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) ≤ | |
| 8*(M:ℝ)*((D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K)) := by | |
| let a := (G.ends e).out.1 | |
| let b := (G.ends e).out.2 | |
| let δ := (G.law p).real (G.twoClusterEvent a b (4^K)) | |
| let Q := (D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K) | |
| have hQ : 0≤Q := by | |
| dsimp [Q] | |
| exact mul_nonneg (div_nonneg (Nat.cast_nonneg _) hp.1.le) (by positivity) | |
| change δ ≤ 8*(M:ℝ)*Q | |
| by_cases hδ : 0<δ | |
| swap | |
| · exact (not_lt.mp hδ).trans (by positivity) | |
| obtain ⟨J,hJ,hJmass⟩ := G.exists_finite_witnessedTwoCluster p a b (4^K) hδ | |
| obtain ⟨W,hW⟩ := G.finite_bonds_in_ball hconnected a (hJ.insert e) | |
| have hJW : J⊆G.inducedEdges (G.vertexBall a W) := fun f hf => hW (Or.inr hf) | |
| have heW : e∈G.inducedEdges (G.vertexBall a W) := hW (Or.inl rfl) | |
| choose φ hφ using hdense a | |
| obtain ⟨n,_,hn⟩ := G.subexponential_ball_enlargement hlocal hg a (C+W) 0 | |
| let B := G.vertexBall a n | |
| let S := G.vertexBall a (n+C+W) | |
| have hBf : B.Finite := G.vertexBall_finite hlocal a n | |
| have hSf : S.Finite := G.vertexBall_finite hlocal a _ | |
| have hEf := G.finite_inducedEdges hlocal hSf | |
| let q (f : E) := (G.law p).real {ω | G.inducedEdges S∩ω ∈ | |
| G.twoClusterEvent (G.ends f).out.1 (G.ends f).out.2 (4^K)} | |
| have hq : ∀ f, 0≤q f := fun _ => measureReal_nonneg | |
| have hmap : ∀ v∈hBf.toFinset, (φ v).edge e∈hEf.toFinset := by | |
| intro v hv | |
| apply hEf.mem_toFinset.mpr | |
| exact G.translated_bonds_in_ball a a v n C W (φ v) (hBf.mem_toFinset.mp hv) | |
| (hφ v) (Set.singleton_subset_iff.mpr heW) ⟨e,rfl,rfl⟩ | |
| have hvbound : ∀ v∈hBf.toFinset, δ/2≤q ((φ v).edge e) := by | |
| intro v hv | |
| have hsub := G.translated_bonds_in_ball a a v n C W (φ v) | |
| (hBf.mem_toFinset.mp hv) (hφ v) hJW | |
| have hm := measureReal_mono (G.witnessedTwoCluster_subset_restrict hsub | |
| ((φ v).vertex a) ((φ v).vertex b) (4^K)) (μ := G.law p) | |
| rw [Aut.witnessedTwoCluster_real G (φ v) p J hJ a b (4^K)] at hm | |
| have hend : s((φ v).vertex a,(φ v).vertex b) = | |
| s((G.ends ((φ v).edge e)).out.1,(G.ends ((φ v).edge e)).out.2) := by | |
| calc | |
| _ = G.ends ((φ v).edge e) := by | |
| rw [(φ v).preserves,(G.ends e).out_eq.symm,Sym2.map_mk] | |
| _ = _ := (G.ends ((φ v).edge e)).out_eq.symm | |
| rw [G.twoClusterEvent_sym2 hend (4^K)] at hm | |
| exact hJmass.trans hm | |
| have hfib := G.orbit_edge_fibre_bound hlocal C M hM a e (Sym2.out_fst_mem _) | |
| φ hφ hBf.toFinset | |
| have havg := G.induced_averaged_adjacent hlocal p hp D hDpos hD S hSf K | |
| change (∑ f∈hEf.toFinset, q f) ≤ (S.ncard:ℝ)*Q at havg | |
| have hratio : S.ncard≤2*B.ncard := by simpa only [S,B,Nat.add_assoc] using hn | |
| have hsum : (B.ncard:ℝ)*(δ/2) ≤ (B.ncard:ℝ)*(4*(M:ℝ)*Q) := by | |
| calc | |
| _ = ∑ v∈hBf.toFinset, δ/2 := by | |
| simp only [Finset.sum_const, nsmul_eq_mul] | |
| rw [Set.ncard_eq_toFinset_card B hBf] | |
| _ ≤ ∑ v∈hBf.toFinset, q ((φ v).edge e) := Finset.sum_le_sum hvbound | |
| _ ≤ (2*M:ℕ)*∑ f∈hEf.toFinset, q f := | |
| sum_comp_le_fibre_bound hBf.toFinset hEf.toFinset (fun v => (φ v).edge e) q | |
| (2*M) hq hmap hfib | |
| _ ≤ (2*M:ℕ)*((S.ncard:ℝ)*Q) := | |
| mul_le_mul_of_nonneg_left havg (Nat.cast_nonneg _) | |
| _ ≤ (2*M:ℕ)*(((2*B.ncard:ℕ):ℝ)*Q) := | |
| mul_le_mul_of_nonneg_left | |
| (mul_le_mul_of_nonneg_right (Nat.cast_le.mpr hratio) hQ) (Nat.cast_nonneg _) | |
| _ = _ := by push_cast; ring | |
| have hBpos : (0:ℝ)<B.ncard := Nat.cast_pos.mpr (G.vertexBall_ncard_pos hlocal a n) | |
| have hh := (mul_le_mul_iff_right₀ hBpos).mp hsum | |
| linarith | |
| theorem subexponential_adjacent_bound (hconnected : G.fullGraph.Connected) | |
| (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) | |
| (hg : G.SubexponentialGrowth) : | |
| ∃ A : ℝ, 0<A ∧ ∀ (p : unitInterval), (p:ℝ)∈Set.Ioo 0 1 → ∀ (e : E) (K : ℕ), | |
| (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) ≤ | |
| A/((p:ℝ)*(2:ℝ)^K) := by | |
| obtain ⟨C,hC⟩ := G.uniform_orbit_density_radius hconnected hquasi | |
| obtain ⟨M,hMpos,hM⟩ := G.uniform_positive_ball_bound hquasi C | |
| obtain ⟨D,hDpos,hD⟩ := G.uniform_incident_bound hquasi | |
| have hm : (0:ℝ)<M := by exact_mod_cast hMpos | |
| have hd : (0:ℝ)<D := by exact_mod_cast hDpos | |
| refine ⟨48*(M:ℝ)*D, by positivity, ?_⟩ | |
| intro p hp e K | |
| have hh := G.adjacent_bound_of_geometry hconnected hlocal hg C M D hM hDpos hD hC p hp e K | |
| convert hh using 1 | |
| ring | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |