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d5b562f | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.MathematicalPhysics.CriticalSK.Main
import OAI.Probability.SKGap.Main
namespace OAI
noncomputable section
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter
namespace CriticalSK
lemma disorderLaw_temperature_map (n : ℕ) (β : ℝ) :
(disorderLaw n).map (fun W => β • W) = SKGap.disorderLaw β n := by
unfold disorderLaw SKGap.disorderLaw
change (Measure.pi (fun _ : Edge n => gaussianReal 0 (n : ℝ≥0)⁻¹)).map
(fun W e => β * W e) = _
rw [Measure.pi_map_pi (f := fun _ : Edge n => fun a : ℝ => β * a)
(fun _ => (measurable_const.mul measurable_id).aemeasurable)]
congr 1
funext e
rw [gaussianReal_map_const_mul, mul_zero]
congr 1
apply NNReal.coe_injective
simp only [NNReal.coe_mul, NNReal.coe_mk, NNReal.coe_inv, NNReal.coe_natCast]
rw [Real.coe_toNNReal _ (div_nonneg (sq_nonneg β) (Nat.cast_nonneg n))]
exact (div_eq_mul_inv _ _).symm
lemma companion_hamiltonian {n : ℕ} (W : Disorder n) (x : Spin n) :
SKGap.hamiltonian W 0 x = hamiltonian W x := by
unfold SKGap.hamiltonian
simp only [Pi.zero_apply, zero_mul, Finset.sum_const_zero, add_zero]
rw [symmetric_sum_upper _ (by
intro i j
rw [SKGap.coupling_symm W i j]
ring)]
have hd : (∑ i : Fin n, SKGap.spinValue (x i) * SKGap.coupling W i i *
SKGap.spinValue (x i)) = 0 := by simp [SKGap.coupling_diag]
rw [hd, zero_add]
have he : (∑ e : Edge n, SKGap.spinValue (x e.val.1) *
SKGap.coupling W e.val.1 e.val.2 * SKGap.spinValue (x e.val.2)) = hamiltonian W x := by
unfold hamiltonian
apply Finset.sum_congr rfl
intro e _
simp only [SKGap.coupling, dite_eq_left e.property]
change spinValue (x e.val.1) * W e * spinValue (x e.val.2) = _
ring
rw [he]
ring
lemma companion_partition {n : ℕ} (W : Disorder n) : SKGap.partition W 0 = partition W := by
simp only [SKGap.partition, SKGap.weight, companion_hamiltonian, partition]
lemma companion_mass {n : ℕ} (W : Disorder n) (x : Spin n) :
SKGap.mass W 0 x = gibbs W x := by
rw [SKGap.mass, SKGap.weight, companion_hamiltonian, companion_partition]
rfl
lemma companion_expectation {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) :
SKGap.expectation W 0 f = mean W f := by
simp only [SKGap.expectation, companion_mass, mean]
lemma companion_variance {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) :
SKGap.variance W 0 f = variance W f := by
simp only [SKGap.variance, companion_expectation, variance]
lemma companion_conditionalExpectation {n : ℕ} (W : Disorder n) (i : Fin n)
(f : Spin n → ℝ) (x : Spin n) :
SKGap.conditionalExpectation W 0 i f x = ∑ y, siteKernel W i x y * f y := by
rw [← siteAverage_eq, siteAverage_two]
rw [← companion_mass W x, show flipSite i x = SKGap.flip i x from rfl,
← companion_mass W (SKGap.flip i x)]
simp only [SKGap.conditionalExpectation, SKGap.mass, div_mul_eq_mul_div, ← add_div,
div_div_div_cancel_right₀ (ne_of_gt (SKGap.partition_pos W 0))]
lemma companion_dirichlet {n : ℕ} (W : Disorder n) (f : Spin n → ℝ) :
SKGap.dirichlet W 0 f = dirichlet W f := by
simp only [SKGap.dirichlet, companion_expectation, companion_conditionalExpectation, dirichlet]
lemma companion_poincare_preimage (n : ℕ) (β C : ℝ) :
(fun W : Disorder n => β • W) ⁻¹' SKGap.poincareEvent n C = poincareEvent n β C := by
ext W
simp only [Set.mem_preimage, SKGap.poincareEvent, poincareEvent, Set.mem_ofPred_eq,
companion_variance, companion_dirichlet]
lemma companion_poincare_measurable (n : ℕ) (C : ℝ) :
MeasurableSet (SKGap.poincareEvent n C) := by
have he : SKGap.poincareEvent n C = poincareEvent n 1 C := by
simpa only [one_smul, Set.preimage_id'] using companion_poincare_preimage n 1 C
rw [he]
exact (poincareEvent_closed n 1 C).measurableSet
lemma companion_poincare_probability (n : ℕ) (β C : ℝ) :
SKGap.disorderLaw β n (SKGap.poincareEvent n C) = disorderLaw n (poincareEvent n β C) := by
rw [← disorderLaw_temperature_map,
Measure.map_apply (continuous_const_smul β).measurable (companion_poincare_measurable n C),
companion_poincare_preimage]
theorem fixed_temperature_poincare (β : ℝ) (hβ0 : 0 < β) (hβ1 : β < 1) :
∃ C : ℝ, 1 ≤ C ∧ Tendsto (fun n : ℕ =>
(disorderLaw n).real (poincareEvent n β C)) atTop (𝓝 1) := by
obtain ⟨C, hC, hP, _⟩ := SKGap.sk_main β hβ0 hβ1
have hp : Tendsto (fun n : ℕ => (disorderLaw n).real (poincareEvent n β C))
atTop (𝓝 1) := by
simp_rw [companion_poincare_probability] at hP
simpa only [Measure.real, ENNReal.toReal_one, Function.comp_def] using
(ENNReal.tendsto_toReal (by simp : (1 : ℝ≥0∞) ≠ ⊤)).comp hP
refine ⟨max 1 C, le_max_left _ _, ?_⟩
apply tendsto_of_tendsto_of_tendsto_of_le_of_le hp tendsto_const_nhds
· intro n
refine measureReal_mono ?_
intro W hW f
exact (hW f).trans (mul_le_mul_of_nonneg_right (le_max_right _ _)
(dirichlet_nonneg (β • W) f))
· intro n
exact measureReal_le_one
end CriticalSK
end
end OAI
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