| import OAI.Probability.ClassicalON.CoarseTail |
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| namespace OAI |
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| noncomputable section |
| open scoped Classical |
| namespace ClassicalON |
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| theorem siteDistance_nonneg (x y : Site) : 0 ≤ siteDistance x y := Real.sqrt_nonneg _ |
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| theorem siteDistance_le_radius_twice (x y : Site) : siteDistance x y ≤ 2*(siteRadius x y:ℝ) := by |
| have hi := siteRadius_nonneg x y |
| have h1 : |(y.1:ℝ)-(x.1:ℝ)| ≤ (siteRadius x y:ℝ) := by |
| exact_mod_cast (le_max_left |y.1-x.1| |y.2-x.2|) |
| have h2 : |(y.2:ℝ)-(x.2:ℝ)| ≤ (siteRadius x y:ℝ) := by |
| exact_mod_cast (le_max_right |y.1-x.1| |y.2-x.2|) |
| have hr : (0:ℝ) ≤ siteRadius x y := by exact_mod_cast hi |
| rw [abs_sub_comm] at h1 h2 |
| have hs1 := mul_self_le_mul_self (abs_nonneg _) h1 |
| have hs2 := mul_self_le_mul_self (abs_nonneg _) h2 |
| have hab1 := sq_abs ((x.1:ℝ)-(y.1:ℝ)) |
| have hab2 := sq_abs ((x.2:ℝ)-(y.2:ℝ)) |
| unfold siteDistance |
| apply Real.sqrt_le_iff.mpr |
| constructor |
| · positivity |
| · nlinarith |
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| theorem half_pow_exp (t : ℕ) : (1/2:ℝ)^t=Real.exp (-Real.log 2*(t:ℝ)) := by |
| rw [mul_comm,Real.exp_nat_mul,Real.exp_neg,Real.exp_log (by norm_num : (0:ℝ) < 2)] |
| norm_num |
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| namespace LatticeGraph |
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| theorem connection_exponential (β : ℝ) (hβ : 0 ≤ β) : |
| ∃ m : ℝ,0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),(∀ e,0 ≤ b e ∧ b e ≤ β) → |
| ∀ x y : G.vertices,G.connectionProbability b x y ≤ 4*Real.exp (-m*siteDistance x.val y.val) := by |
| obtain ⟨N,hN,ht⟩ := connection_geometric_tail β hβ |
| have hNr : (0:ℝ) < N := by exact_mod_cast hN |
| have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| let m := Real.log 2/(50*(N:ℝ)) |
| refine ⟨m,by positivity,?_⟩ |
| intro G b hb x y |
| let f := siteDistance x.val y.val/(50*(N:ℝ)) |
| have hf : 0 ≤ f := div_nonneg (siteDistance_nonneg _ _) (by positivity) |
| let t : ℕ := Nat.floor f-1 |
| have hft : f < (t:ℝ)+2 := by |
| have h1 := Nat.lt_floor_add_one f |
| have h2 : Nat.floor f ≤ t+1 := by dsimp [t]; omega |
| have h3 : (Nat.floor f:ℝ) ≤ (t:ℝ)+1 := by exact_mod_cast h2 |
| linarith |
| have hp : G.connectionProbability b x y ≤ (1/2:ℝ)^t := by |
| by_cases ht0 : t=0 |
| · simpa only [ht0,pow_zero] using G.connectionProbability_le_one b (fun e => (hb e).1) x y |
| · apply ht G b hb x y t |
| have he : Nat.floor f=t+1 := by dsimp [t] at *; omega |
| have hfl := Nat.floor_le hf |
| rw [he,Nat.cast_add,Nat.cast_one] at hfl |
| have hmul := (le_div_iff₀ (show (0:ℝ) < 50*(N:ℝ) by positivity)).mp hfl |
| have hd := siteDistance_le_radius_twice x.val y.val |
| have hr : (N:ℝ)*((25*t:ℕ)+4) ≤ (siteRadius x.val y.val:ℝ) := by |
| push_cast |
| nlinarith |
| exact_mod_cast hr |
| apply le_trans hp |
| rw [half_pow_exp] |
| calc |
| _ ≤ Real.exp (Real.log 2*(2-f)) := Real.exp_le_exp.mpr (by nlinarith) |
| _=4*Real.exp (-m*siteDistance x.val y.val) := by |
| have he : Real.exp (Real.log 2*2)=(4:ℝ) := by |
| rw [mul_comm,show (2:ℝ)=(2:ℕ) by norm_num,Real.exp_nat_mul] |
| norm_num [Real.exp_log] |
| rw [show Real.log 2*(2-f)=Real.log 2*2+(-Real.log 2*f) by ring,Real.exp_add,he] |
| congr 2 |
| dsimp [f,m] |
| ring |
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| end LatticeGraph |
| end ClassicalON |
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| end |
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| end OAI |
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