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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import OAI.Probability.FKPressure.Polydisk
namespace OAI
noncomputable section
open Finset FormalMultilinearSeries
open scoped Classical Topology NNReal ENNReal
namespace BelowOne.FKPressure
open PressureSeries
variable {V I : Type*} [Fintype V]
def complexSitePolynomial (ends : I → V × V) (E : Finset I)
(s : ℝ) (v : I → ℝ) (x : V → ℂ) : ℂ :=
(∑ A ∈ E.powerset, (∏ e ∈ A, (v e : ℂ)) *
∏ c : (graph ⊥ ends A).ConnectedComponent, (1 + (s : ℂ) *
∏ i ∈ univ.filter (fun i => (graph ⊥ ends A).connectedComponentMk i = c), x i)) /
∏ e ∈ E, (1 + (v e : ℂ))
lemma complexSitePolynomial_real (ends : I → V × V) (E : Finset I)
(s : ℝ) (v : I → ℝ) (x : V → ℝ) :
complexSitePolynomial ends E s v (fun i => (x i : ℂ)) =
(percolationSitePolynomial ends E s v x : ℂ) := by
unfold complexSitePolynomial percolationSitePolynomial componentProduct
push_cast
rfl
lemma analyticAt_complexSitePolynomial (ends : I → V × V) (E : Finset I)
(s : ℝ) (v : I → ℝ) (x : V → ℂ) :
AnalyticAt ℂ (complexSitePolynomial ends E s v) x := by
unfold complexSitePolynomial
apply AnalyticAt.div_const
apply Finset.analyticAt_fun_sum
intro A hA
apply AnalyticAt.mul analyticAt_const
apply Finset.analyticAt_fun_prod
intro c hc
apply AnalyticAt.add analyticAt_const
apply AnalyticAt.mul analyticAt_const
apply Finset.analyticAt_fun_prod
intro i hi
exact (ContinuousLinearMap.proj i : (V → ℂ) →L[ℂ] ℂ).analyticAt x
lemma complex_pressure_exponential (ends : I → V × V) (E : Finset I)
{u : ℝ} (hu : 0 < u) (hu1 : u < 1) (v : I → ℝ) (hv : ∀ e ∈ E, 0 ≤ v e)
{r : ℝ} (hr : 1 < r)
(hR : ENNReal.ofReal r < (PositiveAnalyticCube.taylor (pressure ends E u v)).radius) :
∀ x : V → ℂ, (∀ i, ‖x i‖ < r) →
Complex.exp (- (complexify (PositiveAnalyticCube.taylor (pressure ends E u v))).sum x) =
complexSitePolynomial ends E (-u) v x := by
let p := PositiveAnalyticCube.taylor (pressure ends E u v)
have hp := pressure_expansion ends E hu hu1 v hv
have hn : ∀ a : Labels V, 0 ≤ weight p a := fun a => hp.2.1 a.1 a.2
have hb {x : V → ℂ} (hx : ∀ i, ‖x i‖ < r) : x ∈ Metric.eball 0 (complexify p).radius := by
apply mem_eball_zero_iff.mpr
have hnorm := norm_lt_of_coordinates (zero_lt_one.trans hr) hx
exact (show ‖x‖ₑ ≤ ENNReal.ofReal r by
simpa only [ofReal_norm] using ENNReal.ofReal_le_ofReal hnorm.le).trans_lt
(hR.trans_le (complexify_radius p hn))
have hf : AnalyticOnNhd ℂ (fun x => Complex.exp (- (complexify p).sum x))
{x | ∀ i, ‖x i‖ < r} := by
intro x hx
exact ((complexify p).analyticOnNhd x (hb hx)).neg.cexp
have hg : AnalyticOnNhd ℂ (complexSitePolynomial ends E (-u) v)
{x | ∀ i, ‖x i‖ < r} := fun x _ => analyticAt_complexSitePolynomial ends E (-u) v x
have he : ∀ t : V → ℝ, (∀ i, t i ∈ Set.Icc 0 1) →
Complex.exp (- (complexify p).sum (fun i => (t i : ℂ))) =
complexSitePolynomial ends E (-u) v (fun i => (t i : ℂ)) := by
intro t ht
have hsum : p.sum t = pressure ends E u v t := (hp.2.2 t ht).tsum_eq
rw [complexify_sum_real, hsum, complexSitePolynomial_real]
unfold pressure
rw [Complex.ofReal_neg, neg_neg, ← Complex.ofReal_exp,
Real.exp_log (percolationSitePolynomial_pos ends E hu.le hu1 v hv t ht)]
exact identity_on_polydisk hr hf hg he
lemma pressure_coefficient_connected (ends : I → V × V) (E : Finset I)
(u : ℝ) (v : I → ℝ) (hv : ∀ e ∈ E, 0 ≤ v e) (m : V → ℕ)
(hm : 0 < coefficient (PositiveAnalyticCube.taylor (pressure ends E u v)) m) :
((graph ⊥ ends E).induce {i | 0 < m i}).Connected := by
obtain ⟨⟨n,τ⟩,ha,hw⟩ := coefficient_pos_has_label
(PositiveAnalyticCube.taylor (pressure ends E u v)) m hm
have hn : 0 < n := by
by_contra! hn
have hzero : n = 0 := Nat.eq_zero_of_le_zero hn
subst n
apply hw
simp only [weight, PositiveAnalyticCube.taylor, _root_.smul_apply,
iteratedFDeriv_zero_apply, pressure_zero ends E u v hv, smul_zero]
have hc := taylor_coordinate_connected hn τ ends E u v hv hw
have hset : {i | 0 < m i} = Set.range τ := by
ext i
rw [← ha]
exact multiplicity_pos ⟨n,τ⟩ i
rwa [hset]
theorem positive_site_pressure (ends : I → V × V) (E : Finset I)
{q : ℝ} (hq : 0 < q) (hq1 : q < 1) (v : I → ℝ)
(hv : ∀ e ∈ E, 0 ≤ v e) :
∃ (c : (V → ℕ) → ℝ) (r : ℝ),
1 < r ∧ c 0 = 0 ∧ (∀ m, 0 ≤ c m) ∧
(∀ m, 0 < c m → ((graph ⊥ ends E).induce {i | 0 < m i}).Connected) ∧
(∀ x : V → ℂ, (∀ i, ‖x i‖ < r) →
Summable (fun m : V → ℕ => ‖(c m : ℂ) * ∏ i, x i ^ m i‖) ∧
Complex.exp (-(∑' m : V → ℕ, (c m : ℂ) * ∏ i, x i ^ m i)) =
complexSitePolynomial ends E (q - 1) v x) ∧
(∀ x : V → ℝ, (∀ i, x i ∈ Set.Icc 0 1) →
HasSum (fun m : V → ℕ => c m * ∏ i, x i ^ m i)
(-Real.log (percolationSitePolynomial ends E (q - 1) v x))) := by
let u := 1 - q
have hu : 0 < u := sub_pos.mpr hq1
have hu1 : u < 1 := by dsimp [u]; linarith
have huq : -u = q - 1 := by dsimp [u]; ring
let p := PositiveAnalyticCube.taylor (pressure ends E u v)
have hp := pressure_expansion ends E hu hu1 v hv
have hn : ∀ a : Labels V, 0 ≤ weight p a := fun a => hp.2.1 a.1 a.2
obtain ⟨r,hr1,hrR⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hp.1
have hr : 1 < (r : ℝ) := by exact_mod_cast hr1
have hR : ENNReal.ofReal (r : ℝ) < p.radius := by simpa only [ENNReal.ofReal_coe_nnreal] using hrR
refine ⟨coefficient p,r,hr,?_,coefficient_nonneg p hn,?_,?_,?_⟩
· apply coefficient_zero
ext w
simp only [p, PositiveAnalyticCube.taylor, _root_.smul_apply,
iteratedFDeriv_zero_apply, pressure_zero ends E u v hv, smul_zero, _root_.zero_apply]
· exact pressure_coefficient_connected ends E u v hv
· intro x hx
have hs := complexify_hasSum p hn (zero_lt_one.trans hr) hR x hx
refine ⟨hs.summable.norm, ?_⟩
rw [hs.tsum_eq]
simpa only [huq] using complex_pressure_exponential ends E hu hu1 v hv hr hR x hx
· intro x hx
have hcx : ∀ i, ‖(x i : ℂ)‖ < (r : ℝ) := fun i => by
rw [Complex.norm_real, Real.norm_of_nonneg (hx i).1]
exact (hx i).2.trans_lt hr
have hs := complexify_hasSum p hn (zero_lt_one.trans hr) hR (fun i => (x i : ℂ)) hcx
have hsum : p.sum x = pressure ends E u v x := (hp.2.2 x hx).tsum_eq
rw [complexify_sum_real, hsum] at hs
apply Complex.hasSum_ofReal.mp
simpa only [pressure, huq, Complex.ofReal_mul, Complex.ofReal_prod, Complex.ofReal_pow] using hs
end BelowOne.FKPressure
end
end OAI
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