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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry
import OAI.Probability.CriticalPercolation.Growth.HomogeneousGrowth
import OAI.Probability.CriticalPercolation.Harmonic.GraphProjectedFunctions
import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkEnergy
namespace OAI
section
noncomputable section
open Set Function Filter
open scoped Classical Pointwise BigOperators Topology
namespace CriticalPercolation.HomogeneousAnalysis.Geometry
variable {A : Type*} (W : Geometry A)
include W in
lemma countable_vertices : Countable A := by
apply Set.countable_univ_iff.mp
have hc : (⋃ n:ℕ, (W.ball n:Set A)).Countable :=
Set.countable_iUnion (fun n => (W.ball n).finite_toSet.countable)
apply hc.mono
intro a _
obtain ⟨n,hn⟩ := W.exists_mem_ball a
exact Set.mem_iUnion.mpr ⟨n,hn⟩
lemma exists_projected_approximation [Infinite A] (d : ℕ)
(hg : W.HasDegree d) (δ : ℝ) (hδ : 0<δ) :
∃ u : A → ℝ, u W.root=0 ∧ (∀ x, |u x|≤(W.length x:ℝ)) ∧
(∀ x y, W.graph.Adj x y → |u y-u x|≤1) ∧
(W.degree:ℝ)⁻¹^2≤∑ s∈W.neighbors W.root, (u s)^2 ∧
∀ x, |∑ s∈W.neighbors x, (u x-u s)|≤δ := by
have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos
obtain ⟨n,hE,hres⟩ := W.exists_walk_small_relative_residual d hg
(δ^2/(W.degree:ℝ)) (div_pos (sq_pos_of_pos hδ) hc)
let f := W.walkState n
obtain ⟨t,ht,hlarge⟩ := W.exists_large_neighbor_gradient W.degree_pos n
let u := W.projectedFunction n t
have hlow := W.projectedFunction_neighbor_lower n t hlarge
refine ⟨u,W.projectedFunction_root n t,W.projectedFunction_length_bound n t ht,
fun x y h => W.projectedFunction_edge_bound n t ht h,?_,?_⟩
· have hs : (W.degree:ℝ)⁻¹^2≤(u t)^2 :=
pow_le_pow_left₀ (inv_nonneg.mpr hc.le) hlow 2
exact hs.trans (Finset.single_le_sum (fun s hs => sq_nonneg (u s)) ht)
· intro x
have h := W.projectedFunction_residual_sq n t ht x
have hb : (W.degree:ℝ)*‖f-W.lazyWalk f‖^2/W.walkEnergy f≤δ^2 := by
apply (div_le_iff₀ hE).mpr
have hh := mul_le_mul_of_nonneg_left hres hc.le
have he : (W.degree:ℝ)*(δ^2/(W.degree:ℝ)*W.walkEnergy f)=δ^2*W.walkEnergy f := by
field_simp
rw [he] at hh
exact hh
have hh : (∑ s∈W.neighbors x, (u x-u s))^2≤δ^2 := h.trans hb
nlinarith [sq_abs (∑ s∈W.neighbors x, (u x-u s)),abs_nonneg (∑ s∈W.neighbors x, (u x-u s))]
theorem exists_nonconstant_lipschitz_harmonic [Infinite A] (d : ℕ)
(hg : W.HasDegree d) :
∃ u : A → ℝ, W.IsHarmonic u ∧ u W.root=0 ∧
(∀ x, |u x|≤(W.length x:ℝ)) ∧
(∀ x y, W.graph.Adj x y → |u y-u x|≤1) ∧
∃ s∈W.neighbors W.root, u s≠0 := by
let _ : Countable A := W.countable_vertices
have hδ (n:ℕ) : (0:ℝ)<1/((n:ℝ)+1) := by positivity
choose v hv1 hvbound hvedge hvgrad hvres using
(fun n : ℕ => W.exists_projected_approximation d hg (1/((n:ℝ)+1)) (hδ n))
let K : Set (A → ℝ) := Set.pi Set.univ (fun x => Set.Icc (-(W.length x:ℝ)) (W.length x:ℝ))
have hK : IsCompact K := isCompact_univ_pi (fun _ => isCompact_Icc)
have hvK (n:ℕ) : v n∈K := by
intro x hx
exact abs_le.mp (hvbound n x)
obtain ⟨u,huK,φ,hφ,hlim⟩ := hK.tendsto_subseq hvK
have hxlim (x:A) : Tendsto (fun n => v (φ n) x) atTop (𝓝 (u x)) :=
(continuous_apply x).continuousAt.tendsto.comp hlim
have hu1 : u W.root=0 := by
have hh : Tendsto (fun n => v (φ n) W.root) atTop (𝓝 0) := by
simpa only [hv1] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
exact tendsto_nhds_unique (hxlim W.root) hh
have hbound (x:A) : |u x|≤(W.length x:ℝ) := abs_le.mpr (huK x (Set.mem_univ x))
have hedge (x y:A) (h:W.graph.Adj x y) : |u y-u x|≤1 :=
le_of_tendsto ((hxlim y).sub (hxlim x)).abs (Filter.Eventually.of_forall (fun n => hvedge (φ n) x y h))
have hgrad : (W.degree:ℝ)⁻¹^2≤∑ s∈W.neighbors W.root, (u s)^2 := by
have hl := tendsto_finsetSum (W.neighbors W.root) (fun s hs => (hxlim s).pow 2)
exact ge_of_tendsto hl (Filter.Eventually.of_forall (fun n => hvgrad (φ n)))
have hharm : W.IsHarmonic u := by
intro x
have hl := tendsto_finsetSum (W.neighbors x) (fun s hs => (hxlim x).sub (hxlim s))
have hz : Tendsto (fun n : ℕ => 1/((φ n:ℝ)+1)) atTop (𝓝 0) :=
(tendsto_one_div_add_atTop_nhds_zero_nat).comp hφ.tendsto_atTop
have hnonneg : (0:ℝ)≤ |∑ s∈W.neighbors x, (u x-u s)| := abs_nonneg _
have hle : |∑ s∈W.neighbors x, (u x-u s)|≤0 :=
le_of_tendsto_of_tendsto hl.abs hz (Filter.Eventually.of_forall (fun n => hvres (φ n) x))
exact abs_eq_zero.mp (le_antisymm hle hnonneg)
refine ⟨u,hharm,hu1,hbound,hedge,?_⟩
by_contra! hn
have hz : (∑ s∈W.neighbors W.root, (u s)^2)=0 := by
apply Finset.sum_eq_zero
intro s hs
rw [hn s hs]
norm_num
rw [hz] at hgrad
have hc : (0:ℝ)<W.degree := by exact_mod_cast W.degree_pos
have hp := sq_pos_of_pos (inv_pos.mpr hc)
linarith
end CriticalPercolation.HomogeneousAnalysis.Geometry
end
end
end OAI
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