| import OAI.Probability.FreeIsing.Estimates.FiniteWeightNormalization | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem367 | |
| theorem transitionKernel_tanh_eq_exp (beta : ℝ) (a b : IsingSpin) : | |
| transitionKernel (Real.tanh beta) a b = | |
| Real.exp (beta * spinSign a * spinSign b) / (2 * Real.cosh beta) := by | |
| have hc : Real.cosh beta ≠ 0 := ne_of_gt (Real.cosh_pos beta) | |
| cases a <;> cases b <;> | |
| simp only [transitionKernel, spinSign_neg, spinSign_pos, mul_neg, mul_one, | |
| neg_neg] <;> | |
| rw [Real.tanh_eq_sinh_div_cosh] <;> | |
| field_simp <;> | |
| nlinarith [Real.cosh_add_sinh beta, Real.cosh_sub_sinh beta] | |
| theorem prod_transitionKernel_tanh_eq_exp | |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) | |
| (eta : TreeVertex d → IsingSpin) : | |
| (∏ e ∈ inducedEdges D, transitionKernel (Real.tanh beta) (eta e.1) (eta e.2)) = | |
| Real.exp (beta * ∑ e ∈ inducedEdges D, spinSign (eta e.1) * spinSign (eta e.2)) / | |
| (2 * Real.cosh beta) ^ (inducedEdges D).card := by | |
| simp_rw [transitionKernel_tanh_eq_exp] | |
| rw [Finset.prod_div_distrib, Finset.prod_const, ← Real.exp_sum] | |
| congr 2 | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro e he | |
| ring | |
| theorem freeCylinderWeight_tanh_eq_exp | |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) | |
| (eta : TreeVertex d → IsingSpin) : | |
| freeCylinderWeight (Real.tanh beta) D eta = ENNReal.ofReal | |
| (Real.exp (beta * ∑ e ∈ inducedEdges D, spinSign (eta e.1) * spinSign (eta e.2)) / | |
| (2 * (2 * Real.cosh beta) ^ (inducedEdges D).card)) := by | |
| rw [freeCylinderWeight_eq_transitionKernel, prod_transitionKernel_tanh_eq_exp] | |
| congr 1 | |
| ring | |
| theorem finiteIsing_partitionFunction | |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) : | |
| (∑ eta : D → IsingSpin, Real.exp (beta * ∑ e ∈ inducedEdges D, | |
| spinSign (extendFiniteSpin D eta e.1) * spinSign (extendFiniteSpin D eta e.2))) = | |
| 2 * (2 * Real.cosh beta) ^ (inducedEdges D).card := by | |
| classical | |
| have hden : 0 < 2 * (2 * Real.cosh beta) ^ (inducedEdges D).card := by | |
| positivity | |
| have hn := sum_freeCylinderWeight_eq_one (Real.neg_one_lt_tanh beta).le | |
| (Real.tanh_lt_one beta).le D hD | |
| simp_rw [freeCylinderWeight_tanh_eq_exp] at hn | |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun eta _ => | |
| div_nonneg (Real.exp_pos _).le hden.le)] at hn | |
| have hreal := ENNReal.ofReal_eq_one.mp hn | |
| rw [← Finset.sum_div] at hreal | |
| exact (div_eq_one_iff_eq (ne_of_gt hden)).mp hreal | |
| theorem finiteFreeIsingPMF_tanh_eq_normalized_exp | |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) | |
| (eta : D → IsingSpin) : | |
| finiteFreeIsingPMF (Real.neg_one_lt_tanh beta).le (Real.tanh_lt_one beta).le D hD eta = | |
| ENNReal.ofReal (Real.exp (beta * ∑ e ∈ inducedEdges D, | |
| spinSign (extendFiniteSpin D eta e.1) * spinSign (extendFiniteSpin D eta e.2)) / | |
| (∑ xi : D → IsingSpin, Real.exp (beta * ∑ e ∈ inducedEdges D, | |
| spinSign (extendFiniteSpin D xi e.1) * spinSign (extendFiniteSpin D xi e.2)))) := by | |
| rw [finiteFreeIsingPMF_apply, freeCylinderWeight_tanh_eq_exp, finiteIsing_partitionFunction beta D hD] | |
| theorem log_transitionKernel_tanh (beta : ℝ) (a b : IsingSpin) : | |
| Real.log (transitionKernel (Real.tanh beta) a b) = | |
| beta * spinSign a * spinSign b - Real.log (2 * Real.cosh beta) := by | |
| rw [transitionKernel_tanh_eq_exp, Real.log_div (ne_of_gt (Real.exp_pos _)) | |
| (ne_of_gt (by positivity : 0 < 2 * Real.cosh beta)), Real.log_exp] | |
| theorem freeCylinderWeight_tanh_pos | |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) | |
| (eta : TreeVertex d → IsingSpin) : | |
| 0 < freeCylinderWeight (Real.tanh beta) D eta := by | |
| rw [freeCylinderWeight_tanh_eq_exp] | |
| exact ENNReal.ofReal_pos.mpr (by positivity) | |
| theorem HasFreeIsingLaw.cylinder_boltzmann | |
| {d : ℕ} {beta : ℝ} {mu : MeasureTheory.Measure (TreeVertex d → IsingSpin)} | |
| (hmu : HasFreeIsingLaw (Real.tanh beta) mu) | |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) | |
| (eta : TreeVertex d → IsingSpin) : | |
| mu {x | ∀ v ∈ D, x v = eta v} = ENNReal.ofReal | |
| (Real.exp (beta * ∑ e ∈ inducedEdges D, spinSign (eta e.1) * spinSign (eta e.2)) / | |
| (2 * (2 * Real.cosh beta) ^ (inducedEdges D).card)) := by | |
| rw [hmu D hD eta, freeCylinderWeight_tanh_eq_exp] | |
| end Problem367 | |
| end | |
| end OAI | |