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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.PolynomialFolner
namespace OAI
section
noncomputable section
open Set Function Filter Finset MeasureTheory
open scoped Classical Topology BigOperators Pointwise ENNReal NNReal BoundedContinuousFunction
namespace CriticalPercolation.SparseGrowth
variable {G X : Type*} [Group G] [DecidableEq G]
variable [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
variable [MulAction G X] [ContinuousConstSMul G X]
lemma orbit_wordBall_nonempty (S : Finset G) (h1 : (1:G)∈S) (n : ℕ) : (S^n).Nonempty :=
⟨1,Finset.one_mem_pow h1⟩
def orbitAverage (T : Finset G) (hT : T.Nonempty) (x : X) : ProbabilityMeasure X :=
⟨((T.card:ℝ≥0∞)⁻¹) • ∑ g∈T, Measure.dirac (g • x), by
constructor
simp only [Measure.smul_apply,Measure.finsetSum_apply,Measure.dirac_apply_of_mem (Set.mem_univ _),
Finset.sum_const,nsmul_eq_mul,smul_eq_mul,mul_one]
exact ENNReal.inv_mul_cancel (by exact_mod_cast (Finset.card_ne_zero.mpr hT)) (by simp)⟩
omit [DecidableEq G] [ContinuousConstSMul G X] in
lemma integral_orbitAverage (T : Finset G) (hT : T.Nonempty) (x : X)
(f : X → ℝ) :
∫ y, f y ∂(orbitAverage T hT x:Measure X) =
(T.card:ℝ)⁻¹ * ∑ g∈T, f (g • x) := by
change ∫ y, f y ∂(((T.card:ℝ≥0∞)⁻¹) • ∑ g∈T, Measure.dirac (g • x))=_
rw [integral_smul_measure,integral_finsetSum_measure (fun g _ => integrable_dirac (by finiteness))]
simp only [integral_dirac,ENNReal.toReal_inv,ENNReal.toReal_natCast,smul_eq_mul]
omit [ContinuousConstSMul G X] in
lemma orbitAverage_translate_bound (S : Finset G) (h1 : (1:G)∈S)
(n : ℕ) (x : X) (s : G) (hs : s∈S) (f : X →ᵇ ℝ) :
|(∫ y, f (s • y) ∂(orbitAverage (S^n) (orbit_wordBall_nonempty S h1 n) x:Measure X))-
(∫ y, f y ∂(orbitAverage (S^n) (orbit_wordBall_nonempty S h1 n) x:Measure X))| ≤
((S^n).card:ℝ)⁻¹ * (2*(((S^(n+1)).card:ℝ)-(S^n).card)*‖f‖) := by
rw [integral_orbitAverage (S^n) _ x (fun y => f (s • y)),
integral_orbitAverage (S^n) _ x f,←mul_sub,abs_mul,
abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _))]
apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr (Nat.cast_nonneg _))
simpa only [mul_smul] using translated_ball_sum_bound S h1 n s hs
(fun g => f (g • x)) ‖f‖ (norm_nonneg _) (fun g => by
simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm (g • x))
lemma exists_invariant_probability_measure_generators [CompactSpace X]
(S : Finset G) (h1 : (1:G)∈S) (d : ℕ) (hg : HasDegree S d) (x : X) :
∃ μ : ProbabilityMeasure X, ∀ s∈S, ∀ f : X →ᵇ ℝ,
(∫ y, f (s • y) ∂(μ:Measure X))=∫ y, f y ∂(μ:Measure X) := by
have hex : ∀ k : ℕ, ∃ n≥k,
((S^(n+1)).card:ℝ)-(S^n).card≤((k:ℝ)+1)⁻¹*(S^n).card := by
intro k
exact small_ball_increment S h1 d hg _ (by positivity) k
choose r hr hsmall using hex
let ν : ℕ → ProbabilityMeasure X := fun k => orbitAverage (S^(r k)) (orbit_wordBall_nonempty S h1 (r k)) x
obtain ⟨μ,_,φ,hφ,hlim⟩ := (isCompact_univ : IsCompact (Set.univ:Set (ProbabilityMeasure X))).tendsto_subseq
(x:=ν) (fun _ => Set.mem_univ _)
refine ⟨μ,?_⟩
intro s hs f
let fs : X →ᵇ ℝ := f.compContinuous ⟨fun y => s • y,continuous_const_smul s⟩
have he0 : Tendsto (fun k : ℕ => ((k:ℝ)+1)⁻¹) atTop (𝓝 0) := by
exact tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop)
have he : Tendsto (fun k => 2*‖f‖*((φ k:ℝ)+1)⁻¹) atTop (𝓝 0) := by
simpa using (he0.comp hφ.tendsto_atTop).const_mul (2*‖f‖)
have hb (k : ℕ) :
|(∫ y, fs y ∂(ν (φ k):Measure X))-(∫ y, f y ∂(ν (φ k):Measure X))|≤
2*‖f‖*((φ k:ℝ)+1)⁻¹ := by
apply (orbitAverage_translate_bound S h1 (r (φ k)) x s hs f).trans
have hc : (0:ℝ)<(S^(r (φ k))).card := by
exact_mod_cast Finset.card_pos.mpr ⟨1,Finset.one_mem_pow h1⟩
have hm := mul_le_mul_of_nonneg_left (hsmall (φ k))
(show 0≤((S^(r (φ k))).card:ℝ)⁻¹ * (2*‖f‖) by positivity)
calc
_ = ((S^(r (φ k))).card:ℝ)⁻¹ * (2*‖f‖) *
(((S^(r (φ k)+1)).card:ℝ)-(S^(r (φ k))).card) := by ring
_ ≤ ((S^(r (φ k))).card:ℝ)⁻¹ * (2*‖f‖) *
(((φ k:ℝ)+1)⁻¹*(S^(r (φ k))).card) := hm
_ = _ := by field_simp
have hz : Tendsto (fun k => (∫ y, fs y ∂(ν (φ k):Measure X))-
(∫ y, f y ∂(ν (φ k):Measure X))) atTop (𝓝 0) :=
squeeze_zero_norm (fun k => by simpa only [Real.norm_eq_abs] using hb k) he
have hfs := (ProbabilityMeasure.continuous_integral_boundedContinuousFunction fs).tendsto μ |>.comp hlim
have hf := (ProbabilityMeasure.continuous_integral_boundedContinuousFunction f).tendsto μ |>.comp hlim
exact sub_eq_zero.mp (tendsto_nhds_unique (hfs.sub hf) hz)
end CriticalPercolation.SparseGrowth
namespace CriticalPercolation.SparseGrowth
variable {G X : Type*} [Group G] [DecidableEq G]
variable [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
variable [MulAction G X] [ContinuousConstSMul G X]
def integralInvariantSubgroup (μ : ProbabilityMeasure X) : Subgroup G where
carrier := {g | ∀ f : X →ᵇ ℝ, (∫ y, f (g • y) ∂(μ:Measure X))=∫ y, f y ∂(μ:Measure X)}
one_mem' := by intro f; simp
mul_mem' := by
intro a b ha hb f
let fa : X →ᵇ ℝ := f.compContinuous ⟨fun y => a • y,continuous_const_smul a⟩
calc
_ = ∫ y, fa (b • y) ∂(μ:Measure X) := by simp only [fa,BoundedContinuousFunction.compContinuous_apply,ContinuousMap.coe_mk,mul_smul]
_ = ∫ y, fa y ∂(μ:Measure X) := hb fa
_ = ∫ y, f y ∂(μ:Measure X) := ha f
inv_mem' := by
intro a ha f
let fi : X →ᵇ ℝ := f.compContinuous ⟨fun y => a⁻¹ • y,continuous_const_smul _⟩
have hh := ha fi
simpa only [fi,BoundedContinuousFunction.compContinuous_apply,ContinuousMap.coe_mk,inv_smul_smul] using hh.symm
lemma exists_invariant_probability_measure [CompactSpace X]
(S : Finset G) (h1 : (1:G)∈S) (hgen : Subgroup.closure (S:Set G)=⊤)
(d : ℕ) (hg : HasDegree S d) (x : X) :
∃ μ : ProbabilityMeasure X, ∀ g : G,
Measure.map (fun y => g • y) (μ:Measure X)=(μ:Measure X) := by
obtain ⟨μ,hμ⟩ := exists_invariant_probability_measure_generators S h1 d hg x
have hle : Subgroup.closure (S:Set G) ≤ integralInvariantSubgroup μ :=
(Subgroup.closure_le _).mpr (fun s hs => hμ s hs)
rw [hgen] at hle
refine ⟨μ,?_⟩
intro g
have hi : ∀ f : X →ᵇ ℝ, (∫ y, f (g • y) ∂(μ:Measure X))=∫ y, f y ∂(μ:Measure X) :=
hle (Subgroup.mem_top g)
let ν : ProbabilityMeasure X := μ.map (fun point => g • point)
have he : ν.toFiniteMeasure=μ.toFiniteMeasure := by
apply FiniteMeasure.ext_of_forall_integral_eq
intro f
change (∫ y, f y ∂(Measure.map (fun y => g • y) (μ:Measure X)))=_
rw [integral_map (continuous_const_smul g).measurable.aemeasurable f.continuous.aestronglyMeasurable]
exact hi f
exact congrArg FiniteMeasure.toMeasure he
end CriticalPercolation.SparseGrowth
end
end
end OAI
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