| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced |
| import OAI.Probability.CriticalPercolation.Modular.ModularClusterMass |
| import OAI.Probability.CriticalPercolation.Gluing.RelayMoves |
| import OAI.Probability.CriticalPercolation.Modular.HeavyHarris |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def weightCeiling (v : V) (K : ℝ≥0∞) : Set V := {x | G.stabilizerWeight v x≤K} |
|
|
| def descendingFanEvent (v : V) (K ε : ℝ≥0∞) (N : ℕ) : Set (Set E) := |
| {ω | ∃ S : Finset V, N≤S.card ∧ ∀ x∈S, |
| x∈G.cluster (ω∩G.inducedEdges (G.weightCeiling v K)) v ∧ |
| G.stabilizerWeight v x<ε} |
|
|
| lemma descendingFanEvent_upper (v : V) (K ε : ℝ≥0∞) (N : ℕ) : |
| IsUpperSet (G.descendingFanEvent v K ε N) := by |
| rintro ω η h ⟨S,hS,hs⟩ |
| exact ⟨S,hS,fun x hx => ⟨G.cluster_mono (Set.inter_subset_inter_left _ h) v |
| (hs x hx).1,(hs x hx).2⟩⟩ |
|
|
| lemma descendingFanEvent_finiteWitness (v : V) (K ε : ℝ≥0∞) (N : ℕ) : |
| BernoulliProduct.HasFiniteOpenWitness (G.descendingFanEvent v K ε N) := by |
| rintro ω ⟨S,hS,hs⟩ |
| obtain ⟨J,hJ,hJω,hconn⟩ := G.finite_subset_cluster_finite_witness S.finite_toSet |
| (show (S : Set V)⊆G.cluster (ω∩G.inducedEdges (G.weightCeiling v K)) v from |
| fun x hx => (hs x hx).1) |
| refine ⟨J,hJ,hJω.trans Set.inter_subset_left,S,hS,?_⟩ |
| intro x hx |
| refine ⟨?_,(hs x hx).2⟩ |
| have he : J∩G.inducedEdges (G.weightCeiling v K)=J := |
| Set.inter_eq_left.mpr (hJω.trans Set.inter_subset_right) |
| rw [he] |
| exact hconn hx |
|
|
| lemma measurableSet_descendingFanEvent [Countable E] (v : V) (K ε : ℝ≥0∞) (N : ℕ) : |
| MeasurableSet (G.descendingFanEvent v K ε N) := |
| BernoulliProduct.measurableSet_finiteWitness _ (G.descendingFanEvent_upper v K ε N) |
| (G.descendingFanEvent_finiteWitness v K ε N) |
|
|
| lemma stabilizerWeight_le_modularMass {S : Set V} {v x : V} (hx : x∈S) : |
| G.stabilizerWeight v x≤G.modularMass v S := by |
| simpa only [ite_eq_left hx,modularMass] using |
| (ENNReal.le_tsum x (f := fun y => if y∈S then G.stabilizerWeight v y else 0)) |
|
|
| lemma finite_mass_low_slice_infinite {S : Set V} {v : V} |
| (hi : S.Infinite) (hm : G.modularMass v S≠∞) (ε : ℝ≥0∞) (hε : 0<ε) : |
| (S∩{x | G.stabilizerWeight v x<ε}).Infinite := by |
| have hf : (S∩{x | ε≤G.stabilizerWeight v x}).Finite := by |
| apply (ENNReal.finite_const_le_of_tsum_ne_top hm hε.ne').subset |
| intro x hx |
| change ε ≤ if x∈S then G.stabilizerWeight v x else 0 |
| rw [ite_eq_left hx.1] |
| exact hx.2 |
| have hh := hi.sdiff hf |
| apply hh.mono |
| intro x hx |
| refine ⟨hx.1,?_⟩ |
| change G.stabilizerWeight v x<ε |
| apply lt_of_not_ge |
| intro he |
| exact hx.2 ⟨hx.1,he⟩ |
|
|
| lemma cluster_eq_with_weightCeiling {ω : Set E} {v : V} {K : ℝ≥0∞} |
| (hK : G.modularMass v (G.cluster ω v)≤K) : |
| G.cluster (ω∩G.inducedEdges (G.weightCeiling v K)) v=G.cluster ω v := by |
| apply Set.Subset.antisymm (G.cluster_mono Set.inter_subset_left v) |
| rintro x ⟨w⟩ |
| apply G.walk_reachable_induced _ ω w |
| intro y hy |
| exact (G.stabilizerWeight_le_modularMass (w.takeUntil y hy).reachable).trans hK |
|
|
| lemma bounded_mass_infinite_subset_fan (v : V) (K ε : ℝ≥0∞) (hK : K≠∞) |
| (hε : 0<ε) (N : ℕ) : |
| G.infiniteEvent v \ G.massTailEvent v v K ⊆ G.descendingFanEvent v K ε N := by |
| rintro ω ⟨hi,hm⟩ |
| have hm' : G.modularMass v (G.cluster ω v)≤K := le_of_not_gt hm |
| have hh := G.finite_mass_low_slice_infinite hi (ne_top_of_le_ne_top hK hm') ε hε |
| obtain ⟨S,hS,hcard⟩ := hh.exists_subset_card_eq N |
| refine ⟨S,hcard.ge,fun x hx => ?_⟩ |
| have hs := hS hx |
| rw [G.cluster_eq_with_weightCeiling hm'] |
| exact hs |
|
|
| lemma Aut.weightCeiling_image (φ : G.Aut) (v : V) (K : ℝ≥0∞) : |
| φ.vertex '' G.weightCeiling v K=G.weightCeiling (φ.vertex v) K := by |
| ext x |
| obtain ⟨y,rfl⟩ := φ.vertex.surjective x |
| simp only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right,weightCeiling, |
| Set.mem_ofPred_eq,φ.stabilizerWeight] |
|
|
| lemma Aut.image_inducedEdges (φ : G.Aut) (S : Set V) : |
| φ.edge '' G.inducedEdges S=G.inducedEdges (φ.vertex '' S) := by |
| ext e |
| obtain ⟨f,rfl⟩ := φ.edge.surjective e |
| simp only [Set.mem_image,φ.edge.injective.eq_iff,exists_eq_right,inducedEdges, |
| Set.mem_ofPred_eq,φ.preserves,Sym2.mem_map] |
| constructor |
| · intro h x hx |
| obtain ⟨y,hy,rfl⟩ := hx |
| exact ⟨y,h y hy,rfl⟩ |
| · intro h y hy |
| obtain ⟨z,hz,he⟩ := h (φ.vertex y) ⟨y,hy,rfl⟩ |
| exact (φ.vertex.injective he) ▸ hz |
|
|
| lemma Aut.descendingFanEvent_preimage (φ : G.Aut) (v : V) (K ε : ℝ≥0∞) (N : ℕ) : |
| (fun ω : Set E => φ.edge '' ω) ⁻¹' G.descendingFanEvent (φ.vertex v) K ε N= |
| G.descendingFanEvent v K ε N := by |
| have he (ω : Set E) : |
| (φ.edge '' ω)∩G.inducedEdges (G.weightCeiling (φ.vertex v) K)= |
| φ.edge '' (ω∩G.inducedEdges (G.weightCeiling v K)) := by |
| rw [Set.image_inter φ.edge.injective,φ.image_inducedEdges,φ.weightCeiling_image] |
| have hr (ω : Set E) (x : V) : |
| φ.vertex x∈G.cluster ((φ.edge '' ω)∩G.inducedEdges (G.weightCeiling (φ.vertex v) K)) |
| (φ.vertex v) ↔ x∈G.cluster (ω∩G.inducedEdges (G.weightCeiling v K)) v := by |
| rw [he] |
| exact (SimpleGraph.Iso.reachable_iff (φ := Aut.openGraphIso G φ _)) |
| ext ω |
| constructor |
| · rintro ⟨S,hS,hs⟩ |
| refine ⟨S.image φ.vertex.symm,?_,?_⟩ |
| · rwa [Finset.card_image_of_injective _ φ.vertex.symm.injective] |
| · intro x hx |
| obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx |
| have h := hs y hy |
| refine ⟨(hr ω (φ.vertex.symm y)).mp (by simpa using h.1),?_⟩ |
| simpa only [←Aut.stabilizerWeight G φ v (φ.vertex.symm y),φ.vertex.apply_symm_apply] using h.2 |
| · rintro ⟨S,hS,hs⟩ |
| refine ⟨S.image φ.vertex,?_,?_⟩ |
| · rwa [Finset.card_image_of_injective _ φ.vertex.injective] |
| · intro x hx |
| obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hx |
| exact ⟨(hr ω y).mpr (hs y hy).1,by simpa only [φ.stabilizerWeight] using (hs y hy).2⟩ |
|
|
| lemma Aut.descendingFanEvent_real [Countable E] (φ : G.Aut) (p : unitInterval) |
| (v : V) (K ε : ℝ≥0∞) (N : ℕ) : |
| (G.law p).real (G.descendingFanEvent (φ.vertex v) K ε N)= |
| (G.law p).real (G.descendingFanEvent v K ε N) := by |
| simp only [measureReal_def] |
| congr 1 |
| conv_lhs => rw [←G.law_map_aut p φ] |
| rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge) |
| (G.measurableSet_descendingFanEvent _ _ _ _),φ.descendingFanEvent_preimage] |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
