| 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 |
| |
