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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 | 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
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