| import Mathlib |
| import OAI.Probability.SKRatio.FiniteChain.DiscreteDistanceNonneg |
| import OAI.Probability.SKRatio.Matrices.GaussianTails |
| import OAI.Probability.SKRatio.Gaussian.Planted |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Real Filter |
| namespace SKRatio.Planted |
| open Calculus |
| attribute [local instance] Classical.propDecidable |
| variable {n : ℕ} |
|
|
| lemma sum_edges_spin (x : Spin n) : |
| 2*∑ e : Edge n, spin x e.1.1*spin x e.1.2 = (∑ i,spin x i)^2-n := by |
| have h (i j : Fin n) : spin x i*coupling (fun _ : Edge n => 1) i j*spin x j = |
| spin x i*spin x j - if i=j then 1 else 0 := by |
| by_cases hij : i=j |
| · subst j |
| simp only [coupling_diag,mul_zero,zero_mul,↓reduceIte] |
| nlinarith [spin_sq x i] |
| · simp only [coupling] |
| split_ifs <;> simp_all only [mul_one,sub_zero] |
| omega |
| have hs := hamiltonian_edges (fun _ : Edge n => 1) x |
| have hss : (∑ i,spin x i)^2 = ∑ i : Fin n, ∑ j : Fin n, spin x i*spin x j := by |
| rw [pow_two,Finset.sum_mul_sum] |
| simp only [hamiltonian,Pi.zero_apply,zero_mul,Finset.sum_const_zero,add_zero] at hs |
| change (1/2:ℝ)*(∑ i,∑ j,spin x i*coupling (fun _ : Edge n => 1) i j*spin x j) = |
| ∑ e : Edge n,spin x e.1.1*1*spin x e.1.2 at hs |
| simp only [h,Finset.sum_sub_distrib,Finset.sum_ite_eq,Finset.mem_univ,↓reduceIte, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] at hs |
| rw [hss] |
| linarith |
|
|
| lemma weight_pair_linear (g : Disorder n) (x y : Spin n) : |
| weight g 0 x*weight g 0 y = exp (∑ e : Edge n, |
| (spin x e.1.1*spin x e.1.2+spin y e.1.1*spin y e.1.2)*g e) := by |
| rw [weight_linear,weight_linear,←exp_add,←Finset.sum_add_distrib] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro e _ |
| change _ = (spinValue (x e.1.1)*spinValue (x e.1.2)+ |
| spinValue (y e.1.1)*spinValue (y e.1.2))*g e |
| ring |
|
|
| lemma weight_pair_integrable (β : ℝ) (x y : Spin n) : |
| Integrable (fun g => weight g 0 x*weight g 0 y) (disorderLaw β n) := by |
| simp_rw [weight_pair_linear] |
| exact integrable_exp_linear _ _ _ |
|
|
| lemma integral_weight_pair (β : ℝ) (x y : Spin n) : |
| (∫ g, weight g 0 x*weight g 0 y ∂disorderLaw β n) = |
| exp (2*normalizingExponent β n + (β^2/n)* |
| ∑ e : Edge n, spin (signProduct x y) e.1.1*spin (signProduct x y) e.1.2) := by |
| simp_rw [weight_pair_linear] |
| rw [disorderLaw,integral_exp_linear] |
| congr 1 |
| have hv : (Real.toNNReal (β^2/(n:ℝ)) : ℝ) = β^2/n := |
| Real.coe_toNNReal _ (div_nonneg (sq_nonneg _) (Nat.cast_nonneg _)) |
| have he (e : Edge n) : (spin x e.1.1*spin x e.1.2+spin y e.1.1*spin y e.1.2)^2 = |
| 2+2*(spin (signProduct x y) e.1.1*spin (signProduct x y) e.1.2) := by |
| rw [add_sq,mul_pow,mul_pow,spin_sq,spin_sq,spin_sq,spin_sq,spin_signProduct,spin_signProduct] |
| ring |
| simp only [mul_zero,zero_add,hv,he, |
| show ∀ a b : ℝ, a*(2+2*b)/2 = a+a*b from by intros; ring, |
| Finset.sum_add_distrib,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,←Finset.mul_sum, |
| normalizingExponent] |
| ring |
|
|
| lemma partition_square_integrable (β : ℝ) (n : ℕ) : |
| Integrable (fun g : Disorder n => (partition g 0)^2) (disorderLaw β n) := by |
| have he (g : Disorder n) : partition g 0^2 = |
| ∑ x : Spin n, ∑ y : Spin n,weight g 0 x*weight g 0 y := by |
| rw [pow_two,partition,Finset.sum_mul_sum] |
| simp_rw [he] |
| exact integrable_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => weight_pair_integrable β x y)) |
|
|
| lemma sign_sum_exp (t : ℝ) (n : ℕ) : |
| (∑ x : Spin n, exp (t*∑ i,spin x i)) = (2*cosh t)^n := by |
| simp_rw [Finset.mul_sum,exp_sum] |
| have he : (∑ x : Spin n, ∏ i, exp (t*spin x i)) = |
| ∏ _i : Fin n, ∑ b : Bool, exp (t*spinValue b) := by |
| simpa only [spin,spinValue] using |
| (Fintype.prod_sum (fun (_ : Fin n) (b : Bool) => exp (t*spinValue b))).symm |
| rw [he] |
| have hB : (∑ b : Bool, exp (t*spinValue b)) = 2*cosh t := by |
| simp [spinValue,cosh_eq] |
| ring |
| simp only [hB,Finset.prod_const,Finset.card_univ,Fintype.card_fin] |
|
|
| lemma sign_square_exp_bound {β : ℝ} (hβ : β^2 ≤ 1/2) (n : ℕ) : |
| (∑ x : Spin n, exp ((β^2/(2*n))*(∑ i,spin x i)^2)) ≤ 2*(2:ℝ)^n := by |
| by_cases hn : n=0 |
| · subst n |
| norm_num [Spin] |
| have hnR : 0 < (n:ℝ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn) |
| let a := β/sqrt n |
| have ha : a^2 = β^2/n := by |
| dsimp [a] |
| rw [div_pow,sq_sqrt hnR.le] |
| have he (x : Spin n) : exp ((β^2/(2*n))*(∑ i,spin x i)^2) = |
| ∫ z : ℝ, exp ((a*∑ i,spin x i)*z) ∂gaussianReal 0 1 := by |
| have hm := mgf_gaussianReal (μ := 0) (v := 1) (X := id) (p := gaussianReal 0 1) |
| (by simp) (a*∑ i,spin x i) |
| simp only [mgf,id_eq,NNReal.coe_one,one_mul] at hm |
| rw [hm,mul_pow,ha] |
| congr 1 |
| ring |
| simp_rw [he] |
| rw [←integral_finsetSum _ (fun x _ => integrable_exp_mul_gaussianReal (a*∑ i,spin x i))] |
| have he' (z : ℝ) : (∑ x : Spin n, exp ((a*∑ i,spin x i)*z)) = (2*cosh (a*z))^n := by |
| have hh (x : Spin n) : (a*∑ i,spin x i)*z = (a*z)*∑ i,spin x i := by ring |
| simp_rw [hh] |
| exact sign_sum_exp _ _ |
| simp_rw [he'] |
| have hb (z : ℝ) : (2*cosh (a*z))^n ≤ (2:ℝ)^n*exp (z^2/4) := by |
| rw [mul_pow] |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| calc |
| cosh (a*z)^n ≤ exp ((a*z)^2/2)^n := pow_le_pow_left₀ (cosh_pos _).le (cosh_le_exp_half_sq _) n |
| _ = exp (β^2*z^2/2) := by |
| rw [←exp_nat_mul,mul_pow,ha] |
| congr 1 |
| field_simp |
| _ ≤ exp (z^2/4) := exp_le_exp.mpr (by nlinarith [mul_le_mul_of_nonneg_right hβ (sq_nonneg z)]) |
| have hi : Integrable (fun z : ℝ => (2*cosh (a*z))^n) (gaussianReal 0 1) := by |
| have hh := integrable_finsetSum Finset.univ |
| (fun (x : Spin n) _ => integrable_exp_mul_gaussianReal (μ := 0) (v := 1) (a*∑ i,spin x i)) |
| simpa only [he'] using hh |
| calc |
| _ ≤ ∫ z : ℝ, (2:ℝ)^n*exp (z^2/4) ∂gaussianReal 0 1 := |
| integral_mono hi (SKRatioGaussian.gaussian_square_exp_integrable.const_mul _) hb |
| _ ≤ (2:ℝ)^n*2 := by |
| rw [integral_const_mul] |
| exact mul_le_mul_of_nonneg_left SKRatioGaussian.gaussian_square_exp_integral (by positivity) |
| _ = _ := mul_comm _ _ |
| |
| lemma partition_second_moment {β : ℝ} (hβ : β^2 ≤ 1/2) (n : ℕ) : |
| (∫ g : Disorder n, (partition g 0)^2 ∂disorderLaw β n) ≤ |
| 2*((2:ℝ)^n*exp (normalizingExponent β n))^2 := by |
| have he (g : Disorder n) : partition g 0^2 = |
| ∑ x : Spin n, ∑ y : Spin n,weight g 0 x*weight g 0 y := by |
| rw [pow_two,partition,Finset.sum_mul_sum] |
| simp_rw [he] |
| rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => weight_pair_integrable β x y))] |
| simp_rw [integral_finsetSum _ (fun y _ => weight_pair_integrable β _ y),integral_weight_pair] |
| let f : Spin n → ℝ := fun z => exp (2*normalizingExponent β n +(β^2/n)* |
| ∑ e : Edge n,spin z e.1.1*spin z e.1.2) |
| have heq (x : Spin n) : (∑ y : Spin n,f (signProduct x y)) = ∑ z : Spin n,f z := |
| Equiv.sum_comp ((signProduct_involutive x).toPerm) f |
| change (∑ x : Spin n, ∑ y : Spin n,f (signProduct x y)) ≤ _ |
| simp only [heq,Finset.sum_const,Finset.card_univ,nsmul_eq_mul, |
| Fintype.card_fun,Fintype.card_bool,Fintype.card_fin,Nat.cast_pow,Nat.cast_ofNat] |
| have hb (z : Spin n) : f z ≤ exp (2*normalizingExponent β n)* |
| exp ((β^2/(2*n))*(∑ i,spin z i)^2) := by |
| dsimp [f] |
| rw [←exp_add] |
| apply exp_le_exp.mpr |
| have hv := div_nonneg (sq_nonneg β) (Nat.cast_nonneg n : (0:ℝ) ≤ n) |
| have hsum := sum_edges_spin z |
| have hle : ∑ e : Edge n,spin z e.1.1*spin z e.1.2 ≤ (∑ i,spin z i)^2/2 := by |
| linarith [show (0:ℝ) ≤ n from Nat.cast_nonneg n] |
| have hh := mul_le_mul_of_nonneg_left hle hv |
| have hid : (β^2/n)*((∑ i,spin z i)^2/2) = (β^2/(2*n))*(∑ i,spin z i)^2 := by ring |
| rw [hid] at hh |
| linarith |
| calc |
| _ ≤ (2:ℝ)^n * (exp (2*normalizingExponent β n)* |
| ∑ z : Spin n, exp ((β^2/(2*n))*(∑ i,spin z i)^2)) := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| rw [Finset.mul_sum] |
| exact Finset.sum_le_sum (fun z _ => hb z) |
| _ ≤ (2:ℝ)^n*(exp (2*normalizingExponent β n)*(2*(2:ℝ)^n)) := |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (sign_square_exp_bound hβ n) |
| (exp_pos _).le) (by positivity) |
| _ = _ := by |
| rw [show (2:ℝ)*normalizingExponent β n = normalizingExponent β n+normalizingExponent β n by ring, |
| exp_add] |
| ring |
| |
| lemma partition_lower_positive_probability {β : ℝ} (hβ : β^2 ≤ 1/2) (n : ℕ) : |
| (1/8:ℝ) ≤ (disorderLaw β n).real |
| {g : Disorder n | ((2:ℝ)^n*exp (normalizingExponent β n))/2 ≤ partition g 0} := by |
| let M := (2:ℝ)^n*exp (normalizingExponent β n) |
| have hM : 0 < M := by dsimp [M]; positivity |
| let A := {g : Disorder n | M/2 ≤ partition g 0} |
| have hA : MeasurableSet A := measurableSet_le measurable_const (continuous_partition 0).measurable |
| let I : Disorder n → ℝ := A.indicator 1 |
| have hI : Integrable I (disorderLaw β n) := (integrable_const (1:ℝ)).indicator hA |
| have hi : (∫ g, I g ∂disorderLaw β n) = (disorderLaw β n).real A := integral_indicator_one hA |
| have hp (g : Disorder n) : partition g 0 ≤ M/2+(partition g 0)^2/(8*M)+2*M*I g := by |
| by_cases hg : g ∈ A |
| · have he : I g=1 := Set.indicator_of_mem hg 1 |
| rw [he,mul_one] |
| have hsq := sq_nonneg (partition g 0-4*M) |
| have hb : partition g 0-2*M ≤ partition g 0^2/(8*M) := by |
| apply (le_div_iff₀ (show 0 < 8*M by positivity)).mpr |
| nlinarith only [hsq] |
| linarith only [hb,hM] |
| · have he : I g=0 := Set.indicator_of_notMem hg 1 |
| rw [he,mul_zero,add_zero] |
| have hz : partition g 0 < M/2 := lt_of_not_ge hg |
| exact hz.le.trans (le_add_of_nonneg_right (div_nonneg (sq_nonneg _) (by positivity))) |
| have hineq := integral_mono (partition_integrable β n) |
| (((integrable_const (M/2)).add ((partition_square_integrable β n).div_const (8*M))).add |
| (hI.const_mul (2*M))) hp |
| simp only [Pi.add_apply] at hineq |
| have hsum : Integrable (fun g : Disorder n => M/2+partition g 0^2/(8*M)) |
| (disorderLaw β n) := (integrable_const _).add ((partition_square_integrable β n).div_const _) |
| rw [integral_add hsum (hI.const_mul (2*M)), |
| integral_add (integrable_const (M/2)) ((partition_square_integrable β n).div_const (8*M)), |
| integral_const,probReal_univ,smul_eq_mul,one_mul,integral_div,integral_const_mul,hi, |
| integral_partition] at hineq |
| have hs := partition_second_moment hβ n |
| change _ ≤ 2*M^2 at hs |
| change M ≤ M/2+_+2*M*(disorderLaw β n).real A at hineq |
| have hdiv : (∫ g : Disorder n, partition g 0^2 ∂disorderLaw β n)/(8*M) ≤ M/4 := by |
| apply (div_le_iff₀ (show 0 < 8*M by positivity)).mpr |
| nlinarith only [hs] |
| change (1/8:ℝ) ≤ (disorderLaw β n).real A |
| nlinarith only [hineq,hdiv,hM] |
| |
| end SKRatio.Planted |
| |
| end |
| end |
| |
| end OAI |
| |
