| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.Criticality.MinimalPair |
| import OAI.Probability.CriticalPercolation.Criticality.WitnessEvents |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma adjacent_log_lower (A ε : ℝ) (hA : 0<A) (hε : 0<ε) |
| (p : unitInterval) (hεp : ε≤p) (s : ℕ) (hs : 1 ≤ s) (e : E) |
| (hbound : ∀ K : ℕ, |
| (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) ≤ |
| A/((p:ℝ)*(2:ℝ)^K)) |
| (hpos : 0<(G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s)) : |
| Real.log s/2-Real.log (2*A/ε) ≤ |
| -Real.log ((G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s)) := by |
| let K := Nat.log 4 s |
| let δ := (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s) |
| have hK : 4^K ≤ s := Nat.pow_log_le_self 4 (by omega) |
| have hu : (s:ℝ) < (4:ℝ)^(K+1) := by |
| exact_mod_cast Nat.lt_pow_succ_log_self (by norm_num : 1<(4:ℕ)) s |
| have hp : 0<(p:ℝ) := hε.trans_le hεp |
| have hm : δ≤A/(ε*(2:ℝ)^K) := by |
| calc _ ≤ (G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) := |
| measureReal_mono (G.twoClusterEvent_antitone _ _ hK) |
| _ ≤ A/((p:ℝ)*(2:ℝ)^K) := hbound K |
| _ ≤ _ := div_le_div_of_nonneg_left hA.le (by positivity) |
| (mul_le_mul_of_nonneg_right hεp (by positivity)) |
| have hl := Real.log_le_log hpos hm |
| rw [Real.log_div hA.ne' (by positivity),Real.log_mul hε.ne' (by positivity),Real.log_pow] at hl |
| have hspos : (0:ℝ)<s := Nat.cast_pos.mpr (by omega) |
| have hls := Real.log_lt_log hspos hu |
| rw [Real.log_pow,Nat.cast_add,Nat.cast_one] at hls |
| have hlog4 : Real.log 4=2*Real.log 2 := by |
| rw [show (4:ℝ)=2^2 by norm_num,Real.log_pow] |
| norm_num |
| rw [hlog4] at hls |
| have heq : Real.log (2*A/ε)=Real.log 2+Real.log A-Real.log ε := by |
| rw [Real.log_div (by positivity) hε.ne',Real.log_mul (by norm_num) hA.ne'] |
| rw [heq] |
| change Real.log δ ≤ _ at hl |
| change Real.log s/2-_ ≤ -Real.log δ |
| linarith |
|
|
| lemma twoClusterEvent_of_adj {u v : V} (h : G.fullGraph.Adj u v) : |
| ∃ e : E, ∀ s, G.twoClusterEvent u v s = |
| G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s := by |
| obtain ⟨⟨e,_,he⟩,_⟩ := h |
| refine ⟨e,fun s => G.twoClusterEvent_sym2 ?_ s⟩ |
| exact he.symm.trans (G.ends e).out_eq.symm |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
