| import OAI.Probability.FKPressure.Polydisk |
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| namespace OAI |
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| noncomputable section |
| open Finset FormalMultilinearSeries |
| open scoped Classical Topology NNReal ENNReal |
| namespace BelowOne.FKPressure |
| open PressureSeries |
| variable {V I : Type*} [Fintype V] |
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| 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)) |
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| 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 |
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| 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 |
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| 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 |
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| 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] |
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| 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 |
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| end BelowOne.FKPressure |
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| end |
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| end OAI |
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