File size: 5,373 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Clusters.ClusterTail
namespace OAI
section
section
open scoped BigOperators
open scoped Classical ENNReal
open Set
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Filter Topology
open scoped Topology Classical
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
namespace CriticalPercolation
def bandScale (L j : ℕ) : ℕ := L ^ (2 ^ j)
lemma bandScale_zero (L : ℕ) : bandScale L 0 = L := by simp [bandScale]
lemma bandScale_succ (L j : ℕ) : bandScale L (j + 1) = bandScale L j ^ 2 := by
simp only [bandScale, pow_succ, pow_mul]
lemma bandScale_ge {L : ℕ} (_hL : 2 ≤ L) (j : ℕ) : L ≤ bandScale L j := by
exact Nat.le_pow (by positivity)
lemma bandScale_tendsto {L : ℕ} (hL : 2 ≤ L) :
Tendsto (bandScale L) atTop atTop := by
exact (tendsto_pow_atTop_atTop_of_one_lt (by omega : 1 < L)).comp
(tendsto_pow_atTop_atTop_of_one_lt (by norm_num : 1 < (2 : ℕ)))
lemma log_bandScale {L : ℕ} (_hL : 2 ≤ L) (j : ℕ) :
Real.log (bandScale L j) = (2 : ℝ) ^ j * Real.log L := by
simp [bandScale, Nat.cast_pow, Real.log_pow]
lemma band_weight {L : ℕ} (hL : 2 ≤ L) (j : ℕ) :
(Real.log (bandScale L j))⁻¹ ^ 2 =
(1 / 4 : ℝ) ^ j * (Real.log L)⁻¹ ^ 2 := by
rw [log_bandScale hL, mul_inv_rev, mul_pow, mul_comm]
congr 1
rw [← inv_pow, ← pow_mul, Nat.mul_comm, pow_mul]
norm_num
lemma band_weight_sum_lt {L : ℕ} (hL : 2 ≤ L) (k : ℕ) :
∑ j ∈ Finset.range k, (Real.log (bandScale L j))⁻¹ ^ 2 <
4 / (3 * (Real.log L) ^ 2) := by
have hlog : 0 < Real.log L := Real.log_pos (by exact_mod_cast (show 1 < L by omega))
simp_rw [band_weight hL]
rw [← Finset.sum_mul]
have hg : ∑ j ∈ Finset.range k, (1/4:ℝ)^j < 4/3 := by
have he := geom_sum_mul (1/4:ℝ) k
have hp := pow_pos (by norm_num : (0:ℝ) < 1/4) k
nlinarith
have hi : 0 < (Real.log L)⁻¹ ^ 2 := sq_pos_of_pos (inv_pos.mpr hlog)
have hm := mul_lt_mul_of_pos_right hg hi
convert hm using 1
simp [div_eq_mul_inv, inv_pow, mul_assoc, mul_comm, mul_left_comm]
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
def clusterBand (o : V) (s : ℕ) : Set (Set E) :=
G.clusterTail o s \ G.clusterTail o (s ^ 2)
lemma real_clusterBand [Countable E] (p : unitInterval) (o : V) (s : ℕ) :
(G.law p).real (G.clusterBand o s) =
(G.law p).real (G.clusterTail o s) - (G.law p).real (G.clusterTail o (s ^ 2)) := by
exact measureReal_sdiff (G.clusterTail_antitone o (Nat.le_pow (by omega)))
(G.measurableSet_clusterTail o (s ^ 2))
lemma sum_real_clusterBand [Countable E] (p : unitInterval) (o : V) (L k : ℕ) :
∑ j ∈ Finset.range k, (G.law p).real (G.clusterBand o (bandScale L j)) =
(G.law p).real (G.clusterTail o L) -
(G.law p).real (G.clusterTail o (bandScale L k)) := by
simp_rw [G.real_clusterBand, ← bandScale_succ]
simpa only [bandScale_zero] using Finset.sum_range_sub'
(fun j => (G.law p).real (G.clusterTail o (bandScale L j))) k
theorem exists_subcritical_clusterBand [Nonempty V]
(hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite)
(hquasi : G.QuasiTransitive) (hp : 0 < G.criticalProbability)
(θ : ℝ) (hθ : 0 < θ)
(hroot : ∀ o, θ ≤ (G.law G.criticalProbability).real (G.infiniteEvent o))
(L : ℕ) (hL : 2 ≤ L) (hlarge : 4 / (3 * (Real.log L)^2) < θ / 4) :
∃ q : unitInterval, (G.criticalProbability : ℝ) / 2 ≤ q ∧
q < G.criticalProbability ∧ ∃ z : V, ∃ s : ℕ, L ≤ s ∧
(Real.log s)⁻¹ ^ 2 ≤ (G.law q).real (G.clusterBand z s) ∧
∀ o, θ / 2 ≤ (G.law q).real (G.clusterTail o s) := by
let := G.countable_vertices hconnected hlocal
let := G.countable_bonds hconnected hlocal
obtain ⟨q, hqhalf, hqpc, hq⟩ := G.subcritical_uniform_clusterTail
hconnected hlocal hquasi G.criticalProbability hp θ hθ hroot L
have hex : ∃ k, ∃ z : V, (G.law q).real (G.clusterTail z (bandScale L k)) < θ / 2 := by
let o : V := Classical.arbitrary V
have ht := (G.tendsto_real_clusterTail_subcritical hqpc o).comp (bandScale_tendsto hL)
obtain ⟨k, hk⟩ := (ht.eventually (gt_mem_nhds (by linarith : (0:ℝ) < θ/2))).exists
exact ⟨k, o, hk⟩
let k := Nat.find hex
obtain ⟨z, hz⟩ := Nat.find_spec hex
have hpre : ∀ j < k, ∀ o : V,
θ / 2 ≤ (G.law q).real (G.clusterTail o (bandScale L j)) := by
intro j hj o
by_contra! h
exact Nat.find_min hex hj ⟨o, h⟩
have hsum : θ / 4 < ∑ j ∈ Finset.range k,
(G.law q).real (G.clusterBand z (bandScale L j)) := by
rw [G.sum_real_clusterBand]
have hqz := hq z
change (G.law q).real (G.clusterTail z (bandScale L k)) < θ / 2 at hz
linarith
have hexj : ∃ j ∈ Finset.range k, (Real.log (bandScale L j))⁻¹ ^ 2 ≤
(G.law q).real (G.clusterBand z (bandScale L j)) := by
by_contra! hj
have hle := Finset.sum_le_sum (s := Finset.range k) (fun j hjk => (hj j hjk).le)
have hlt := (band_weight_sum_lt hL k).trans hlarge
linarith
obtain ⟨j, hj, hband⟩ := hexj
exact ⟨q, hqhalf, hqpc, z, bandScale L j, bandScale_ge hL j, hband,
hpre j (Finset.mem_range.mp hj)⟩
end BondGraph
end CriticalPercolation
end
end
end OAI
|