File size: 6,314 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Harmonic.LazyWalkHilbert
import OAI.Probability.CriticalPercolation.Harmonic.WalkEnergyLower
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise BigOperators
namespace CriticalPercolation.HarmonicAnalysis.WordGeometry
variable {A : Type*} [Group A] (W : WordGeometry A)
def projectedFunction (f : RegularHilbert A) (t : A) (x : A) : ℝ :=
inner ℝ (regular x f-f) (regular t f-f) / (4*(W.generators.card:ℝ)*W.walkEnergy f)
lemma exists_large_generator_gradient (f : RegularHilbert A) :
∃ t∈W.generators, 4*W.walkEnergy f≤‖regular t f-f‖^2 := by
by_contra! h
have hh := Finset.sum_lt_sum_of_nonempty (show W.generators.Nonempty from ⟨1,W.one_mem⟩)
(fun s hs => h s hs)
rw [W.walkEnergy_gradient] at hh
simp only [Finset.sum_const,nsmul_eq_mul] at hh
nlinarith
lemma projectedFunction_one (f : RegularHilbert A) (t : A) :
W.projectedFunction f t 1=0 := by simp [projectedFunction]
lemma projectedFunction_edge_bound (f : RegularHilbert A) (hf : 0<W.walkEnergy f)
(t : A) (ht : t∈W.generators) (x s : A) (hs : s∈W.generators) :
|W.projectedFunction f t (x*s)-W.projectedFunction f t x|≤1 := by
let q := 4*(W.generators.card:ℝ)*W.walkEnergy f
have hc : (0:ℝ)<W.generators.card := by exact_mod_cast W.generators_card_pos
have hq : 0<q := mul_pos (mul_pos (by norm_num) hc) hf
have hg (a : A) (ha : a∈W.generators) : ‖regular a f-f‖≤Real.sqrt q := by
have h := W.regular_energy_displacement f 1 (by simpa [ball] using ha)
simpa only [Nat.cast_one,one_mul] using h
have he : (regular (x*s) f-f)-(regular x f-f)=regular x (regular s f-f) := by
rw [regular_mul,map_sub]
abel
unfold projectedFunction
rw [←sub_div,←inner_sub_left,he,abs_div,abs_of_pos hq]
apply (div_le_one hq).mpr
calc
_ ≤ ‖regular x (regular s f-f)‖*‖regular t f-f‖ := abs_real_inner_le_norm _ _
_ = ‖regular s f-f‖*‖regular t f-f‖ := by rw [LinearIsometryEquiv.norm_map]
_ ≤ Real.sqrt q*Real.sqrt q := mul_le_mul (hg s hs) (hg t ht) (norm_nonneg _) (Real.sqrt_nonneg _)
_ = q := Real.mul_self_sqrt hq.le
lemma projectedFunction_length_bound (f : RegularHilbert A) (hf : 0<W.walkEnergy f)
(t : A) (ht : t∈W.generators) (x : A) :
|W.projectedFunction f t x|≤(W.length x:ℝ) := by
let q := 4*(W.generators.card:ℝ)*W.walkEnergy f
have hc : (0:ℝ)<W.generators.card := by exact_mod_cast W.generators_card_pos
have hq : 0<q := mul_pos (mul_pos (by norm_num) hc) hf
have hx := W.regular_energy_displacement f (W.length x) (W.mem_ball_length x)
have ht' := W.regular_energy_displacement f 1 (by simpa [ball] using ht)
simp only [Nat.cast_one,one_mul] at ht'
unfold projectedFunction
rw [abs_div,abs_of_pos hq]
apply (div_le_iff₀ hq).mpr
calc
_ ≤ ‖regular x f-f‖*‖regular t f-f‖ := abs_real_inner_le_norm _ _
_ ≤ ((W.length x:ℝ)*Real.sqrt q)*Real.sqrt q :=
mul_le_mul hx ht' (norm_nonneg _) (by positivity)
_ = (W.length x:ℝ)*q := by rw [mul_assoc,Real.mul_self_sqrt hq.le]
lemma projectedFunction_generator_lower (f : RegularHilbert A) (hf : 0<W.walkEnergy f)
(t : A) (ht : 4*W.walkEnergy f≤‖regular t f-f‖^2) :
(W.generators.card:ℝ)⁻¹≤W.projectedFunction f t t := by
have hc : (0:ℝ)<W.generators.card := by exact_mod_cast W.generators_card_pos
have hq : 0<4*(W.generators.card:ℝ)*W.walkEnergy f := by positivity
unfold projectedFunction
rw [real_inner_self_eq_norm_sq]
apply (le_div_iff₀ hq).mpr
have he : (W.generators.card:ℝ)⁻¹*(4*(W.generators.card:ℝ)*W.walkEnergy f)=4*W.walkEnergy f := by
field_simp
rw [he]
exact ht
lemma sum_walk_displacement (f : RegularHilbert A) :
(∑ s∈W.generators, (f-regular s f))=(2*(W.generators.card:ℝ)) • (f-W.lazyWalk f) := by
have hc : (W.generators.card:ℝ)≠0 := by exact_mod_cast W.generators_card_pos.ne'
rw [Finset.sum_sub_distrib,Finset.sum_const,←Nat.cast_smul_eq_nsmul ℝ,W.lazyWalk_apply,W.walkAverage_apply]
simp only [smul_sub,smul_add,smul_smul]
have h1 : (2*(W.generators.card:ℝ))*(1/2)=W.generators.card := by ring
have h2 : (2*(W.generators.card:ℝ))*(1/2)*(W.generators.card:ℝ)⁻¹=1 := by field_simp
rw [h1,←mul_assoc (2*(W.generators.card:ℝ)) (1/2) (W.generators.card:ℝ)⁻¹,h2,one_smul]
module
lemma projectedFunction_residual (f : RegularHilbert A) (t x : A) :
(∑ s∈W.generators, (W.projectedFunction f t x-W.projectedFunction f t (x*s)))=
(2*(W.generators.card:ℝ))* inner ℝ (regular x (f-W.lazyWalk f)) (regular t f-f) /
(4*(W.generators.card:ℝ)*W.walkEnergy f) := by
simp only [projectedFunction,←sub_div,←Finset.sum_div,←inner_sub_left,←sum_inner]
have he : (∑ s∈W.generators, (regular x f-f-(regular (x*s) f-f)))=
regular x (∑ s∈W.generators, (f-regular s f)) := by
rw [map_sum]
apply Finset.sum_congr rfl
intro s hs
rw [regular_mul,map_sub]
abel
rw [he,W.sum_walk_displacement,map_smul,real_inner_smul_left]
lemma projectedFunction_residual_sq (f : RegularHilbert A) (hf : 0<W.walkEnergy f)
(t : A) (ht : t∈W.generators) (x : A) :
(∑ s∈W.generators, (W.projectedFunction f t x-W.projectedFunction f t (x*s)))^2≤
(W.generators.card:ℝ)*‖f-W.lazyWalk f‖^2/W.walkEnergy f := by
have hc : (0:ℝ)<W.generators.card := by exact_mod_cast W.generators_card_pos
have hgrad := (Finset.single_le_sum (f := fun s => ‖regular s f-f‖^2)
(fun s hs => sq_nonneg _) ht).trans_eq (W.walkEnergy_gradient f)
have hi := abs_real_inner_le_norm (regular x (f-W.lazyWalk f)) (regular t f-f)
rw [LinearIsometryEquiv.norm_map] at hi
have hisq := pow_le_pow_left₀ (abs_nonneg _) hi 2
rw [sq_abs,mul_pow] at hisq
have hb := mul_le_mul_of_nonneg_left hgrad (sq_nonneg ‖f-W.lazyWalk f‖)
rw [W.projectedFunction_residual]
rw [div_pow]
apply (div_le_div_iff₀ (sq_pos_of_pos (show 0<4*(W.generators.card:ℝ)*W.walkEnergy f by positivity)) hf).mpr
have hm := mul_le_mul_of_nonneg_left (hisq.trans hb)
(show 0≤4*(W.generators.card:ℝ)^2*W.walkEnergy f by positivity)
nlinarith only [hm]
end CriticalPercolation.HarmonicAnalysis.WordGeometry
end
end
end OAI
|