| import Mathlib |
| import OAI.Probability.SKGap.Terminal.PairGibbsBound |
| import OAI.Probability.SKGap.Entropy.SparseWeightedForm |
| import OAI.Probability.SKGap.Terminal.SmallFieldCount |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real |
| open scoped BigOperators RealInnerProductSpace Matrix.Norms.L2Operator |
| variable {n : ℕ} |
|
|
| lemma one_sub_tanh_sq (x : ℝ) : 1-tanh x^2=1/cosh x^2 := by |
| rw [tanh_eq_sinh_div_cosh] |
| have hc := (cosh_pos x).ne' |
| have he := cosh_sq_sub_sinh_sq x |
| field_simp |
| nlinarith only [he] |
|
|
| lemma variance_tanh_le_cosh {t x : ℝ} (ht : 0 ≤ t) (hx : t ≤ |x|) : |
| 1-tanh x^2 ≤ 1/cosh t^2 := by |
| rw [one_sub_tanh_sq] |
| have hcosh : cosh t ≤ cosh x := cosh_le_cosh.mpr (by simpa only [abs_of_nonneg ht] using hx) |
| have hs := (sq_le_sq₀ (cosh_pos t).le (cosh_pos x).le).mpr hcosh |
| exact one_div_le_one_div_of_le (sq_pos_of_pos (cosh_pos t)) hs |
|
|
| def terminalU (g : Disorder n) : Matrix (Fin n) (Fin n) ℝ := Matrix.of (fun i j=>tanh (coupling g i j)) |
| def terminalQ (g : Disorder n) : Matrix (Fin n) (Fin n) ℝ := Matrix.of (fun i j=>tanh (coupling g i j)^2) |
|
|
| theorem terminal_deterministic (g : Disorder n) (h : Fin n→ℝ) {M t α ε R : ℝ} |
| (hM : 0 ≤ M) (ht : 0 < t) (hε : 0 ≤ ε) |
| (hJ : ‖Matrix.of (coupling g)‖ ≤ M) |
| (hU : ‖terminalU g‖ ≤ M) (hQ : ‖terminalQ g‖ ≤ M) |
| (hc : ∀ i j,|tanh (coupling g i j)| ≤ 1/1000) |
| (hR : ∀ i,∑ j,tanh (coupling g i j)^4 ≤ R) |
| (hsparse : ∀ S : Finset (Fin n),(S.card:ℝ) ≤ α*(n:ℝ) → |
| ∀ w : EuclideanSpace ℝ (Fin n),(∀ i,i ∉ S → w i=0) → |
| |⟪w,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) (terminalU g) w⟫| ≤ ε*‖w‖^2 ∧ |
| |⟪w,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) (terminalQ g) w⟫| ≤ ε*‖w‖^2) |
| (hB : ((Finset.univ.filter (fun i=>|h i| < 2*t)).card:ℝ) ≤ α*(n:ℝ)/2) |
| (hsmall : M^2 ≤ t^2*α/2) |
| (hnum : 800000*R+400001*(ε+M*(2*sqrt (1/cosh t^2)+1/cosh t^2)) ≤ 1/2) : |
| ∀ f : Spin n→ℝ,variance g h f ≤ 2*dirichlet g h f := by |
| have hη : (0:ℝ) ≤ 1/cosh t^2 := by positivity |
| have hS := all_configurations_small_fields g h hM ht hJ hB hsmall |
| let k := ε+M*(2*sqrt (1/cosh t^2)+1/cosh t^2) |
| have hcurv (f : Spin n→ℝ) : (1/2:ℝ)*dirichlet g h f ≤ |
| expectation g h (fun x=>heatBathGenerator g h f x^2) := by |
| let q := fun x=>∑ i,siteVariance g h i x*spinGradient i f x^2 |
| have hq : expectation g h q=dirichlet g h f := by |
| dsimp [q] |
| rw [expectation_sum,dirichlet_weighted] |
| have hform (x : Spin n) : |
| (∑ i,∑ j,tanh (coupling g i j)*weightedGradient g h i f x*weightedGradient g h j f x) ≤ k*q x ∧ |
| (∑ i,∑ j,tanh (coupling g i j)^2*|weightedGradient g h i f x*weightedGradient g h j f x|) ≤ k*q x := by |
| let S := Finset.univ.filter (fun i=>|localField g h i x| < t) |
| have hv (i : Fin n) : 0 ≤ siteVariance g h i x := (siteVariance_pos g h i x).le |
| have hvsmall (i : Fin n) (hi : i ∉ S) : siteVariance g h i x ≤ 1/cosh t^2 := by |
| apply variance_tanh_le_cosh ht.le |
| have hi' : ¬|localField g h i x| < t := by simpa only [S,Finset.mem_filter,Finset.mem_univ,true_and] using hi |
| exact le_of_not_gt hi' |
| have hboundU := weighted_sparse_form (terminalU g) S (fun i=>siteVariance g h i x) |
| (fun i=>spinGradient i f x) hη hε hM hv (fun i=>siteVariance_le_one g h i x) hvsmall hU |
| (fun w hw=>(hsparse S (hS x) w hw).1) |
| have hboundQ := weighted_sparse_form (terminalQ g) S (fun i=>siteVariance g h i x) |
| (fun i=>|spinGradient i f x|) hη hε hM hv (fun i=>siteVariance_le_one g h i x) hvsmall hQ |
| (fun w hw=>(hsparse S (hS x) w hw).2) |
| constructor |
| · exact (le_abs_self _).trans hboundU |
| · simp only [sq_abs] at hboundQ |
| have he : (∑ i,∑ j,tanh (coupling g i j)^2*|weightedGradient g h i f x*weightedGradient g h j f x|)= |
| ∑ i,∑ j,terminalQ g i j*(siteVariance g h i x*|spinGradient i f x|)* |
| (siteVariance g h j x*|spinGradient j f x|) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [terminalQ,Matrix.of_apply,weightedGradient,abs_mul,abs_of_nonneg (hv i),abs_of_nonneg (hv j)] |
| ring |
| rw [he] |
| exact (le_abs_self _).trans hboundQ |
| have hu := expectation_mono g h (fun x=>(hform x).1) |
| have hq' := expectation_mono g h (fun x=>(hform x).2) |
| rw [expectation_const_mul,hq] at hu hq' |
| have hl := gibbs_signed_curvature g h f hc hR |
| have hd := dirichlet_nonneg g h f |
| have hn : 800000*R+400001*k ≤ 1/2 := hnum |
| have hh := mul_le_mul_of_nonneg_right hn hd |
| nlinarith only [hu,hq',hl,hh] |
| intro f |
| have hp := heatBath_curvature_poincare g h (by norm_num : (0:ℝ) < 1/2) hcurv f |
| linarith only [hp] |
| end SKGap |
|
|
| end |
| end |
|
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| end OAI |
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