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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Exploration.OpenFrontierDivergence
import OAI.Probability.CriticalPercolation.Exploration.SliceExploration
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
import OAI.Probability.CriticalPercolation.Gluing.StripMoves
import OAI.Probability.CriticalPercolation.Modular.ModularClusterMass
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 modularUpperSlice (o : V) (ε : ℝ) : Set V := {x | ε ≤ G.modularRatio x o}
def modularSliceBonds (o : V) (ε : ℝ) : Set E := G.inducedEdges (G.modularUpperSlice o ε)
lemma finite_lightUpperSlice (o v : V) (ε : ℝ) (hε : 0<ε) (ω : Set E)
(hm : G.modularMass o (G.cluster ω v) ≠ ∞) :
(G.cluster ω v ∩ G.modularUpperSlice o ε).Finite := by
apply (ENNReal.finite_const_le_of_tsum_ne_top hm
(ε := ENNReal.ofReal ε) (ENNReal.ofReal_pos.mpr hε).ne').subset
intro x hx
change ENNReal.ofReal ε ≤ if x∈G.cluster ω v then G.stabilizerWeight o x else 0
rw [ite_eq_left hx.1]
exact ENNReal.ofReal_le_ofReal hx.2
lemma finite_lightSliceComponent (o v : V) (ε : ℝ) (hε : 0<ε)
(hv : v∈G.modularUpperSlice o ε) (ω : Set E)
(hm : G.modularMass o (G.cluster ω v) ≠ ∞) :
(G.cluster (ω∩G.modularSliceBonds o ε) v).Finite := by
apply (G.finite_lightUpperSlice o v ε hε ω hm).subset
intro x hx
exact ⟨G.cluster_mono Set.inter_subset_left v hx,
G.cluster_induced_subset ω _ hv hx⟩
lemma modularSliceBonds_exhaust (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o : V) (ε : ℕ → ℝ) (hε : Tendsto ε atTop (𝓝 0)) :
∀ S : Finset E, ∀ᶠ n in atTop, (S : Set E) ⊆ G.modularSliceBonds o (ε n) := by
have hv (x : V) : ∀ᶠ n in atTop, ε n ≤ G.modularRatio x o :=
(hε.eventually (gt_mem_nhds (G.modularRatio_pos hc hl x o))).mono (fun _ h => h.le)
have he (t : Sym2 V) : ∀ᶠ n in atTop, ∀ x∈t, ε n ≤ G.modularRatio x o := by
refine Sym2.inductionOn t ?_
intro x y
filter_upwards [hv x,hv y] with n hn hn'
intro z hz
simp only [Sym2.mem_iff] at hz
rcases hz with rfl|rfl
· exact hn
· exact hn'
intro S
apply S.eventually_all.mpr
intro e _
exact he (G.ends e)
theorem lightSlice_fewOpen_probability_tendsto_zero [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(p : unitInterval) (hp : 0<(p:ℝ)) (hp1 : (p:ℝ)<1) (o v : V)
(ε : ℕ → ℝ) (hεpos : ∀ n, 0<ε n) (hε : Tendsto ε atTop (𝓝 0)) (N : ℕ) :
Tendsto (fun n => G.law p (G.lightInfiniteEvent o v ∩
{ω | (G.freshClusterBoundary (G.modularSliceBonds o (ε n)) v ω ∩ ω).ncard ≤ N}))
atTop (𝓝 0) := by
have hlim := G.fewOpenFreshBoundary_probability_tendsto_zero hl p hp hp1
(fun n => G.modularSliceBonds o (ε n)) (G.modularSliceBonds_exhaust hc hl o ε hε) v N
have hv : ∀ᶠ n in atTop, v∈G.modularUpperSlice o (ε n) :=
(hε.eventually (gt_mem_nhds (G.modularRatio_pos hc hl v o))).mono (fun _ h => h.le)
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
(Eventually.of_forall (fun _ => zero_le))
filter_upwards [hv] with n hn
apply measure_mono
rintro ω ⟨⟨hi,hm⟩,hcount⟩
exact ⟨hi,G.finite_lightSliceComponent o v (ε n) (hεpos n) hn ω hm.ne,hcount⟩
end CriticalPercolation.BondGraph
end
end
end OAI
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