The explicit formulae for scaling limits in the ergodic decomposition of

arXiv:1402.5230v1 [math.FA] 21 Feb 2014

THE EXPLICIT FORMULAE FOR SCALING LIMITS IN THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES ALEXANDER I. BUFETOV , YANQI QIU A BSTRACT. The main result of this paper, Theorem 1.1, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on spaces of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.

1. I NTRODUCTION . 1.1. Outline of the main results. 1.1.1. Pickrell measures. We start by recalling the definition of Pickrell measures [11]. Our presentation follows [6]. (s) Given a parameter s ∈ R and a natural number n, consider a measure µn on the space Mat(n, C) of n × n-complex matrices, given by the formula (1)

∗ −2n−s µ(s) dz. n = constn,s det(1 + z z)

Here dz is the Lebesgue measure on the space of matrices, and constn,s a normalization constant whose choice will be explained later. Note that the (s) measure µn is finite if s > −1 and infinite if s ≤ −1. If the constants constn,s are chosen appropriately, then the sequence of measures (1) has the Kolmogorov property of consistency under natural (s) projections: the push-forward of the measure µn+1 under the natural projection of cutting the n × n-corner of a (n + 1) × (n + 1)-matrix is pre(s) cisely the measure µn . This consistency property also holds for infinite measures provided n is sufficiently large. The consistency property and the Kolmogorov Existence Theorem allows one to define the Pickrell measure µ(s) on the space of infinite complex matrices Mat(N, C), which is finite if s > −1 and infinite if s ≤ −1. Let U(∞) be the infinite unitary group [ U(n), U(∞) = n∈N

1

2

ALEXANDER I. BUFETOV , YANQI QIU

and let G = U(∞) × U(∞). Groups like U(∞) or G are considered as nice “big groups”, they are non-locally compact groups, but are the inductive limits of compact ones. The space Mat(N, C) can be naturally considered as a G-space given by the action Tu1 ,u2 (z) = u1 zu∗2 , for (u1 , u2) ∈ G, z ∈ Mat(N, C). By definition, the Pickrell measures are G-invariant. The ergodic decomposition of Pickrell measures with respect to G-action was studied in [2] in finite case and [6] in infinite case. The ergodic G-invariant probability measures on Mat(N, C) admit an explicit classification due to Pickrell [11] and to which Olshanski and Vershik [10] gave a different approach: let Merg (Mat(N, C)) be the set of ergodic probability measures and define the Pickrell set by ( ) ∞ X ΩP = ω = (γ, x) : x = (x1 ≥ x2 ≥ · · · ≥ xi ≥ · · · ≥ 0), xi ≤ γ , i=1

then there is a natural identification: ΩP ↔ Merg (Mat(N, C)) . ω ↔ ηω Set Ω0P :=

(

ω = (γ, x) ∈ ΩP : xi > 0 for all i, and γ =

∞ X i=1

xi

)

.

The finite Pickrell measures µ(s) admit the following unique ergodic decomposition Z (s) µ = ηω dµ(s) (ω). (2) ΩP

Borodin and Olshanski [2] proved that the decomposition measures µ(s) live on Ω0P , i.e., µ(s) (ΩP \ Ω0P ) = 0. Let B(s) denote the push-forward of the following map: conf : Ω0P → Conf((0, ∞)) . ω 7→ {x1 , x2 , . . . , xi , . . . } The above µ(s) -almost sure bijection identifies the decomposition measure µ(s) on ΩP with the measure B(s) on Conf((0, ∞)), for this reason, the measure B(s) will also be called the decomposition measure of the Pickrell measure µ(s) . It is showed that B(s) is a determinantal measure on Conf((0, ∞))

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

with correlation kernel (3)

J

(s)

1 (x1 , x2 ) = x1 x2

Z

0

1

3

 r   r  t t Js 2 dt. Js 2 x1 x2

The change of variable y = 4/x reduces the kernel J (s) to the well-known kernel Je(s) of the Bessel point process of Tracy and Widom in [16]: Z 1 √ √ 1 (s) Je (x1 , x2 ) = Js ( tx1 )Js ( tx2 )dt. 4 0 When s ≤ −1, the ergodic decomposition of the infinite Pickrell measure (s) µ was described in [6], the decomposition formula takes the same form as (2), while this time, the decomposition measure µ(s) is an infinite measure on ΩP and again, we have µ(s) (ΩP \Ω0P ) = 0. The µ(s) -almost sure bijection ω → conf(ω) identifies µ(s) with an infinite determinantal measure B(s) on Conf((0, ∞)). One suitable way to describe B(s) is via the multiplicative functionals, for which we recall the definition: a multiplicative functional on Conf((0, ∞)) is obtained by taking the product of the values of a fixed nonnegative function over all particles of a configuration: Y Ψg (X) = g(x) for any X ∈ Conf((0, ∞)). x∈X

If the function g : (0, ∞) → (0, 1) is suitably chosen, then

(4)

Ψg B(s) Ψ dB(s) Conf((0,∞)) g

R

is a determinantal measure on Conf((0, ∞)) whose correlation kernel coincides with that of an orthogonal projection Πg : L2 (0, ∞) → L2 (0, ∞). Note that the range Ran(Πg ) of this projection is explicitly given in [6]. However, even for simple g, the explicit formula for the kernel of Πg turns out to be non-trivial. Our aim in this paper is to give explicit formulae for the kernel of the operator Πg for suitable g. The kernels are obtained as the scaling limits of the Christoffel-Darboux kernels associated to ChristoffelUvarov deformation of Jacobi orthogonal polynomial ensembles. 1.1.2. Formulation of the main result. Let f1 , · · · , fn be complex-valued functions on an interval admitting n−1 derivatives. We write W (f1 , . . . , fn ) for the Wronskian of f1 , . . . , fn , which, we recall, is defined by the formula f1 (t) f ′ (t) · · · f (n−1) (t) 1 1 f (t) f ′ (t) · · · f (n−1) (t) 2 2 2 W (f1 , · · · , fn )(t) = . . .. .. .. .. . . . (n−1) f (t) f ′ (t) · · · f (t) n n n

4

ALEXANDER I. BUFETOV , YANQI QIU

For s′ > −1, we write

√ def Js′,y (t) = Js′ (t y),

√ def Ks′ ,vj (t) = Ks′ (t vj ),

where Js′ stands for the Bessel function, Ks′ for the modified Bessel function of the second kind. The main result of this paper is given by the following Theorem 1.1. Let s ≤ −1 and let m be any natural number such that s + m > −1. Assume that v1 , . . . , vm are distinct positive real numbers. Then for the function (5)

m Y

m

Y 4/x 4 g(x) = = , 4/x + vj 4 + vj x j=1 j=1

the kernel Πg is given by the formula (s+m,v) A (1, 4/x) B (s+m,v) (1, 4/x) A(s+m,v) (1, 4/x′) B (s+m,v) (1, 4/x′ ) 1 Πg (x, x′ ) = · Qm p , 2 (vj + 4/x)(vj + 4/x′ ) · [C (s+m,v) (1)]2 · (x′ − x) j=1

where

A(s+m,v) (t, y) = W (Ks+m,v1 , . . . , Ks+m,vm , Js+m,y )(t), B (s+m,v) (t, y) =

∂A(s+m,v) (t, y), ∂t

C (s+m,v) (t) = W (Ks+m,v1 , . . . , Ks+m,vm )(t). Remark 1.2. When s > −1, the above theorem still holds for any m ≥ 1. In this case, for the same g as given in (5), by results of [6], the kernel Πg obtained above is the kernel for the operator of othogonal projection from √ L2 (R+ , Leb) onto the subspace gRanJ (s) (here, with a slight abuse of notation, we let J (s) be the operator of orthogonal projection with kernel given in (3)). Even in this case, however, the only way we can derive the explicit formula, given above, for the kernel Πg is by using the method of scaling limits. 1.2. Organization of the paper. The remainder of the paper is organized as follows. Section 2 is devoted to some preliminary Mehler-Heine type asymptotics for Jacobi polynomials, these asymptotics will be used in the explicit calculations of the scaling limits in section 4. In Section 3, we show that, for three kinds of auxiliary functions g, the scaling limits of the Christoffel-Darboux kernels for the Christoffel-Uvarov

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

5

deformations of Jacobi orthogonal polynomial ensembles coincide with the kernels Πg which generate the determinantal probability given in (4). In section 4, we continue the study of the three kinds of auxiliary functions g. In case I, we illustrate how we calculate the scaling limits, the obtained scaling limits are the kernels for the determinantal process which are deformations of the Bessel point process of Tracy and Widom. The main formulae in Theorem 1.1 will follow from the formulae obtained in case II, given in Theorem 4.18 after change of variables z → 4/x. Acknowledgements. Grigori Olshanski posed the problem to us, and we are greatly indebted to him. We are deeply grateful to Alexei M. Borodin, who suggested to use the Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles. We are deeply grateful to Alexei Klimenko for useful discussions. The authors are supported by A*MIDEX project (No. ANR-11-IDEX0001-02), financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency (ANR). A. B. is also supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by the ANR under the project ”VALET” of the Programme JCJC SIMI 1, and by the RFBR grants 11-01-00654, 12-01-31284, 12-01-33020, 13-01-12449 . Y. Q. is supported in part by the ANR grant 2011-BS01-00801. 2. P RELIMINARY

ASYMPTOTIC FORMULAE .

2.1. Notation. If A, B are two quantities depending on the same variables, we write A A ≍ B if there exist two absolute values c1 , c2 > 0 such that c1 ≤ B ≤ c2 . When A and B positive quantities, we write A . B, if there exists an absolute value c > 0 such that A ≤ cB. For α, β > −1, we denote the Jacobi weight on (−1, 1) by wα,β (t) = (1 − t)α (1 + t)β .

(α,β)

The associated Jacobi polynomials are denoted by Pn . The leading coR (α,β) (α,β) (α,β) (α,β) efficient of Pn is denoted by kn and hn := [Pn (t)]2 wα,β (t)dt. When α = s, β = 0, we will always omit β in the notation: so ws,0 will (s,0) (s) (s,0;ℓ) be denoted by ws , Pn will be denoted by Pn and the quantity ∆Q,n (s;ℓ) defined in the sequel will be denoted by ∆Q,n , etc. (α,β) Given a sequence (fn )∞ n=0 of functions depending on α, β, we define the differences of the sequence by (α,β; 0)

∆f,n

:= fn(α,β) ,

(α,β; ℓ+1)

and for ℓ ≥ 0, ∆f,n

(α,β; ℓ)

(α,β; ℓ)

:= ∆f,n+1 − ∆f,n

.

6

ALEXANDER I. BUFETOV , YANQI QIU (α,β; −1)

By convention, we set ∆f,n ≡ 0. In what follows, κn always stands for a sequence of natural numbers such that κn lim = κ > 0. n→∞ n Typical such sequences are given by κn = ⌊κn⌋. 2.2. Asymptotics for Higher Differences of Jacobi Polynomials. In this section, we establish some asymptotic formulae for higher differences of (α,β; ℓ) Jacobi polynomials ∆P,n . Lemma 2.1. For ℓ ≥ 0 and n ≥ 1, we have (α,β; ℓ+1)

(n + 1)∆P,n

(α,β; ℓ)

(α+1,β;ℓ−1)

(x) + ℓ∆P,n+1 (x) + ℓ(1 − x)∆P,n+1 (x) (6) α+β (α+1,β;ℓ) (α,β; ℓ) +(n + + 1)(1 − x)∆P,n (x) = α∆P,n (x). 2 Proof. When ℓ = 0, identity (6) is reduced to known formula (cf. [15, 4.5.4]): α+β + 1)(1 − x)Pn(α+1,β) (x) (n + 2 (7) (α,β) =(n + 1)(Pn(α,β) (x) − Pn+1 (x)) + αPn(α,β) (x).

Now assume that identity (6) holds for an integer ℓ and for all n ≥ 1. In particular, substituting n + 1 for n, we have (α,β; ℓ+1)

(α,β; ℓ)

(α+1,β;ℓ−1)

(n + 2)∆P,n+1 (x) + ℓ∆P,n+2 (x) + ℓ(1 − x)∆P,n+2 (x) (8) α+β (α+1,β;ℓ) (α,β; ℓ) + 2)(1 − x)∆P,n+1 (x) = α∆P,n+1 (x). + (n + 2 Then (8) − (6) yields that (α,β; ℓ+2)

(α,β; ℓ+1)

(α+1,β;ℓ)

(n + 1)∆P,n (x) + (ℓ + 1)∆P,n+1 (x) + (ℓ + 1)(1 − x)∆P,n+1 (x) α+β (α+1,β;ℓ+1) (α,β; ℓ+1) + 1)(1 − x)∆P,n (x) = α∆P,n (x). + (n + 2 Thus (6) holds for ℓ + 1 and all n ≥ 1. By induction, identity (6) holds for all ℓ ≥ 0 and all n ≥ 1.  The classical Mehler-Heine theorem ([15, p.192]) says that for z ∈ C \ {0},  √ α z  lim n−α Pn(α,β) 1 − 2 = 2α z − 2 Jα ( z). (9) n→∞ 2n This formula holds uniformly for z in a simply connected compact subset of C \ {0}. Applying the above asymptotics, we have

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

Proposition 2.2. In the regime x(n) = 1 − (10)

7

z , 2n2

for ℓ ≥ 0, we have √ ℓ−α (α,β; ℓ) lim nℓ−α ∆P,κn (x(n) ) = 2α z 2 Jα(ℓ) (κ z).

n→∞

The formula holds uniformly in κ and z as long as κ ranges in a compact subset of (0, ∞) and z ranges in a compact simply connected subset of C \ {0}. Proof. When ℓ = 0, identity (10) is readily reduced to the Mehler-Heine asymptotic formula (9) and the uniform convergence. Now assume identity (10) holds for 0, 1, · · · , ℓ, then by (6), we have (11)

(α,β; ℓ+1)

lim nℓ+1−α ∆P,kn

n→∞

(x(n) )

ℓ−1−(α+1) ℓ α ℓ−α (ℓ) √ ℓ z (ℓ−1) √ 2 · 2 z 2 Jα (κ z) − · 2α+1 z Jα+1 (κ z) κ κ 2 √ ℓ−α α z α+1 ℓ−(α+1) (ℓ) √ − 2 z 2 Jα+1 (κ z) + 2α z 2 Jα(ℓ) (κ z) 2 κ (ℓ−1) √ (ℓ) √ (ℓ) √ h √ Jα+1 (κ z) Jα (κ z) i Jα (κ z) (ℓ) α ℓ+1−α √ √ √ =2 z 2 . −ℓ· − Jα+1 (κ z) + α −ℓ· κ z κ z κ z

=−

From the known recurrence relation (cf. [1, 9.1.27]) α (12) Jα′ (z) = −Jα+1 (z) + Jα (z), z by induction on ℓ, one readily sees that, for all ℓ ≥ 1, h i (ℓ) (ℓ−1) (13) z Jα(ℓ+1) (z) + Jα+1 (z) = (α − ℓ)Jα(ℓ) (z) − ℓJα+1 (z).

Identity (10) for ℓ + 1 follows from (11) and (13), thus the proposition is proved by induction on ℓ.  We will also need the asymptotics for the derivative of the differences of Jacobi polynomials. The derivative of the Jacobi polynomials can be expressed in Jacobi polynomials with different parameters, more precisely, we have 1 d n (α,β) o (α+1,β+1) (t) = (n + α + β + 1)Pn−1 Pn P˙ n(α,β) (t) = (14) (t). dt 2 Using this relation, we have Proposition 2.3. In the regime x(n) = 1 −

z , 2n2

for ℓ ≥ 0, we have √ (ℓ) ˙ (α,β; ℓ) (x(n) ) = 2α z −2+ℓ−α 2 Jeα+1 (κ z), lim n−2+ℓ−α ∆ P,κn

n→∞

8

ALEXANDER I. BUFETOV , YANQI QIU

where Jeα+1 (t) := tJα+1 (t). The formula holds uniformly in κ and z as long as κ ranges in a compact subset of (0, ∞) and z ranges in a compact simply connected subset of C \ {0}. Proof. The relation (14) can be written as (α,β; 0)

˙ 2∆ P,n

(α+1,β+1;0)

= (n + α + β + 1)∆P,n−1

.

From this formula, it is readily to deduce by induction that for all ℓ ≥ 0, (15)

(α,β; ℓ)

˙ 2∆ P,n

(α+1,β+1;ℓ)

= (n + α + β + 1)∆P,n−1

(α+1,β+1;ℓ−1)

+ ℓ · ∆P,n

.

In view of Proposition 2.2 and identity (15), we have ˙ (α,β; ℓ) (x(n) ) lim n−2+ℓ−α ∆ P,n h i √ √ (ℓ) (ℓ−1) √ α −2+ℓ−α 2 κ zJα+1 (κ z) + ℓJα+1 (κ z) =2 z √ −2+ℓ−α (ℓ) =2α z 2 Jeα+1 (κ z). n→∞

The last equality follows from Leibniz formula  (ℓ) (ℓ) (ℓ−1) tJα+1 (t) = tJα+1 (t) + ℓJα+1 (t).



2.3. Asymptotics for Higher Differences of Jacobi’s Functions of the (α,β) Second Kind. Let Qn be the Jacobi’s functions of second kind defined as follows. For x ∈ C \ [−1, 1], Z 1 (α,β) 1 (t) (α,β) −α −β α β Pn Qn (x) := (x − 1) (x + 1) dt. (1 − t) (1 + t) 2 x−t −1 w 2n2

with w > 0. Then √ s − 2s Ks (κ w), lim n−s Q(s) κn (1 + rn ) = 2 w

Proposition 2.4. Let s > −1 and rn = n→∞

where Ks is the modified Bessel function of second kind with order s. For any ε > 0, the convergence is uniform as long as κ ∈ [ε, 1] and w ranges in a bounded simply connected subset of C \ {0}. Proof. We show the proposition when κn = n, the general case is similar. Define tn by the formula 1 1 1 + rn = tn + , |tn | < 1. 2 tn By definition, we have √ lim n(1 − tn ) = w. n→∞

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

9

We now use the integral representation for the Jacobi function of the second kind (cf. [15, 4.82.4]). Write Z −s 1  4tn s ∞  (s) (1 + tn )eτ + 1 − tn × Qn (1 + rn ) = 2 1 − tn −∞  −n−1 1 × 1 + rn + (2rn + rn2 ) 2 cosh τ dτ.

Taking n → ∞ and using the integral representation for the modified Bessel function(cf. [1, 9.6.24]), we see that Z ∞ √ −s (s) s−1 − 2s lim n Qn (1 + rn ) = 2 w e−sτ − w cosh τ dτ n→∞ −∞ Z ∞ √ s = 2s−1 w − 2 e− w cosh τ cosh(sτ )dτ Z ∞−∞ √ √ s s = 2s w − 2 e− w cosh τ cosh(sτ )dτ = 2s w − 2 Ks ( w). 0



Proposition 2.5. In the same condition as in Proposition 2.4, we have for all ℓ ≥ 0, √ ℓ−s (s; ℓ) lim nℓ−s ∆Q,κn (1 + rn ) = 2s w 2 Ks(ℓ) (κ w), (16) n→∞

(ℓ)

where Ks is the ℓ-th derivative of the modified Bessel function of second kind Ks . Moreover, For any ε > 0, the convergence is uniform as long as κ ∈ [ε, 1] and w ranges in a bounded simply connected subset of C \ {0}. Proof. It suffices to prove the proposition in the case κn = n. The general case can easily be deduced from this special case by using the uniform convergence. From the identity (7) we obtain s (s; 1) (s+1; 0) (s; 0) (n + 1)∆Q,n (x) + (n + + 1)(x − 1)∆Q,n (x) = s∆Q,n (x). 2 By induction, it is readily to write (17)

(s; ℓ+1)

(s; ℓ)

(s+1; ℓ−1)

(n + 1)∆Q,n (x) + ℓ∆Q,n+1 (x) + ℓ(x − 1)∆Q,n+1 s (s+1; ℓ) (s; ℓ) +(n + + 1)(x − 1)∆Q,n (x) = s∆Q,n (x), 2 (s; −1)

for all ℓ ≥ 0 where by convention, we set ∆Q,n := 0 . Using the formula ([1, 9.6.26]) s (18) Ks′ (t) = −Ks+1 (t) + Ks (t), t

(x)

10

ALEXANDER I. BUFETOV , YANQI QIU

we can show that for ℓ ≥ 1, h i (ℓ) (ℓ−1) (ℓ+1) (19) t Ks (t) + Ks+1 (t) = (s − ℓ)Ks(ℓ) (t) − ℓKs+1 (t).

Proposition 2.4 says that the equation (16) holds for ℓ = 0. Now assume (16) holds for 0, 1, · · · , ℓ. By (17), we have (s; ℓ+1)

lim nℓ+1−s ∆Q,n

n→∞

(n)

(1 + rj )

ℓ−s−2 ℓ−s wj √ (ℓ−1) √ = −ℓ · 2s wj 2 Ks(ℓ) ( wj ) − ℓ · 2s+1 wj 2 Ks+1 ( wj ) 2 ℓ−s wj s+1 ℓ−s−1 √ √ (ℓ) − · 2 wj 2 Ks+1 ( wj ) + s · 2s wj 2 Ks(ℓ) ( wj ) 2 h i ℓ−s √ (ℓ−1) √ (ℓ) √ (ℓ) √ s 2 = 2 wj (s − ℓ)Ks ( wj ) − ℓKs+1 ( wj ) − wj Ks+1 ( wj ) ℓ+1−s 2

= 2s wj

√ Ks(ℓ+1) ( wj ).



This completes the proof. (α,β)

2.4. Asymptotics for Higher Differences of Rn

.

Definition 2.6. Define for x ∈ C \ [−1, 1], Z 1 (α,β) Pn (t) (α,β) −α −β Rn (x) := (x − 1) (x + 1) (1 − t)α (1 + t)β dt. 2 −1 (x − t) Definition 2.7. For any s ∈ R, define Ls (x) := sKs (x) −

xKs−1 (x) + xKs+1 (x) . 2

Proposition 2.8. Let s > −1, and γ (n) =

u 2n2

lim n−2−s Rκ(s)n (1 + γ (n) ) = 2

2s+3 2

n→∞

with u > 0. Then we have √ s+2 · u− 2 Ls (κ u).

Moreover, for any ε > 0, the convergence is uniform as long as κ ∈ [ε, 1] and u ranges in a compact subset of (0, ∞). Proof. The uniform convergence can be derived by a careful look at the following proof. By this uniform convergence, it suffices to show the proposition for κn = n. Define z by the formula 1 1 z+ , |z| < 1. x= 2 z

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

11

By the integral representation for the Jacobi function of the second kind ([15, 4.82.4]), we have Z −s 1  4z s ∞  (s) Qn (x) = (1 + z)eτ + 1 − z × 2 1−z −∞ −n−1  1 dτ. × x + (x2 − 1) 2 cosh τ Denote

b(s) (x) := 2(x − 1)s Q(s) (x) = Q n n

Then (20)



We have

Z

1 −1

(s)

Pn (t) (1 − t)s dt. x−t

 Z 1 (s) Pn (t) d b(s) Qn (x) = − (1 − t)s dt = −(x − 1)s Rn(s) (x). 2 dx −1 (x − t)

b(s) (x) = 2s Q n

Z

∞

−∞

Hence

where (n) T1 (x)

1+ 

r

x+1 τ e x−1

!−s



1

x + (x2 − 1) 2 cosh τ

 d b(s) (n) (n) (x) = T1 (x) − T2 (x), Q dx n

s · 2s = (x − 1)2

r

x−1 x+1

Z

∞

eτ

−∞

1+ 

r

x+1 τ e x−1 2

!−s−1 1 2

(n) T2 (x)

= (n + 1)2s

We have s−2

lim n

n→∞

dτ.

×

× x + (x − 1) cosh τ and

−n−1

−n−1

dτ

!−s  −n−2 1 x+1 τ x + (x2 − 1) 2 cosh τ e × 1+ x−1 −∞ x × (1 + √ cosh τ )dτ. x2 − 1

Z

r

∞

(n) T1 (1

+γ

(n)

√

s−2 2

Z

∞

2s · u e−sτ − −∞ √ √ s−2 = 2 2s · u 2 Ks ( u).

)=

√

u cosh τ

dτ

12

ALEXANDER I. BUFETOV , YANQI QIU s−2

lim n

n→∞

(n) T2 (1

+γ

(n)

)= =

Hence

√ √

2u 2u

s−1 2

s−1 2

Z



∞

e−sτ e−

√

u cosh τ

cosh τ dτ

−∞

√ √
Ks+1 ( u) + Ks−1 ( u) .

 d b(s) lim n Qn (1 + γ (n) ) n→∞ dx  √ √ √  √ √ s−2 √ uKs+1 ( u) + uKs−1 ( u) 2 =2 2u sKs ( u) − 2 √ s−2 √ =2 2u 2 Ls ( u). s−2



In view of (20), we prove the desired result. Remark. We have the following relations (21) (22)

s L′s (x) = −Ls+1 (x) + Ls (x), x h i (ℓ) (ℓ−1) x Ls(ℓ+1) (x) + Ls+1 (x) = (s − ℓ)L(ℓ) s (x) − ℓLs+1 (x).

Let us for example show (21). The validity of (22) can be verified easily by induction on ℓ. We have ′ ′ xKs−1 (x) + Ks−1 (x) + xKs+1 (x) + Ks+1 (x) L′s (x) =sKs′ (x) − 2 2 s −xKs (x) + (s − 1)Ks−1(x) = − sKs+1 (x) + Ks (x) − x 2 −xKs+2 (x) + (s + 1)Ks+1(x) − 2   xKs (x) + xKs+2 (x) = − (s + 1)Ks+1(x) − 2   xKs−1 (x) + xKs+1 (x) s sKs (x) − + x 2 s = − Ls+1 (x) + Ls (x). x Proposition 2.9. Let s > −1, and γ (n) = we have (s; ℓ)

lim nℓ−s−2 ∆R,κn (1 + γ (n) ) = 2

n→∞

u 2n2 2s+3 2

with u > 0. Then for ℓ ≥ 0, ·u

ℓ−s−2 2

√ L(ℓ) s (κ u).

Moreover, for any ε > 0, the convergence is uniform as long as κ ∈ [ε, 1] and u ranges in a compact subset of (0, ∞).

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

13

Proof. Again, we show the proposition only for κn = n. The formula holds for ℓ = 0. Assume that the formula holds for all 0, 1, · · · , ℓ, we shall show (s; ℓ) that it holds for ℓ + 1. By similar arguments as that for ∆Q,n , we can easily obtain that (s; ℓ+1)

(s; ℓ)

(s+1; ℓ−1)

(n + 1)∆R,n (x) + ℓ∆R,n+1 (x) + ℓ(x − 1)∆R,n+1 s (s+1; ℓ) (s; ℓ) + (n + + 1)(x − 1)∆R,n (x) = s∆R,n (x). 2

(x)

Hence (s; ℓ+1)

lim n(ℓ+1)−s−2 ∆R,n

(1 + γ (n) ) √ ℓ−s−2 ℓ−s−4 2s+3 u 2s+5 (ℓ−1) √ 2 · u 2 Ls+1 ( u) =(s − ℓ)2 2 · u 2 L(ℓ) s ( u) − ℓ · 2 2 ℓ−s−3 u 2s+5 (ℓ) √ − · 2 2 · u 2 Ls+1 ( u) 2   2s+3 ℓ−s−1 s − ℓ (ℓ) √ ℓ (ℓ−1) √ (ℓ) √ 2 2 √ Ls ( u) − √ Ls+1 ( u) − Ls+1 ( u) =2 ·u u u √ (ℓ+1)−s−2 2s+3 (ℓ+1) Ls+1 ( u). =2 2 · u 2 n→∞



3. B ESSEL P OINT P ROCESSES AS R ADIAL PARTS OF P ICKRELL M EASURES ON I NFINITE M ATRICES 3.0.1. Radial parts of Pickrell measures and the infinite Bessel point processes. Following Pickrell, we introduce a map by the formula

radn : Mat(n, C) → Rn+

radn (z) = (λ1 (z ∗ z), . . . , λn (z ∗ z)). Here (λ1 (z ∗ z), . . . , λn (z ∗ z)) is the collection of the eigenvalues of the matrix z ∗ z arranged in non-decreasing order. (s) (s) The radial part of the Pickrell measure µn is defined as (radn )∗ µn . Note that, since finite-dimensional unitary groups are compact, even for s ≤ −1, (s) if n + s > 0, then the radial part of µn is well-defined. Denote dλ the Lebesgue measure on Rn+ , then the radial part of the mea(s) sure µn takes the form Y 1 dλ. constn,s · (λi − λj )2 · (1 + λi )2n+s i −1, (n) II. g (n) = gII and g = gII with m + s > −1, (n) III. g (n) = gIII and g = gIII with m + s > −1. Then the determinantal probability measure in (26) converges weakly in Mfin(Conf((0, +∞))) to (34)

Ψg B(s) . Ψ dB(s) Conf((0,∞)) g

R P(s) g :=

Proof. We will use the notation in [6], where, following [2], it is proved that the measure µ(s) is supported on the subset Matreg (N, C) for any s ∈ R. By the remarks preceding the proposition, for any z ∈ Matreg (N, C), we have lim Ψeg(n) (r(n) (z)) = Ψg (r∞ (z)),

n→∞

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

17

and Ψeg(n) (r(n) (z)) ≤ M · Ψg (r(n) (z)). Now take any bounded and continuous function f on Conf((0, ∞)), we have R Z f (r(n) (z))Ψeg(n) (r(n) (z))dµ(s) (z) Matreg (N,C) (s,n) R f (X)dPeg(n) (X) = . Ψg(n) (r(n) (z))dµ(s) (z) Matreg (N,C) e

By Lemma 3.1, it suffices to show that Z Z (n) (s) (35) lim Ψg (r (z))dµ (z) = n→∞

Mat(N,C)

Ψg (r(∞) (z))dµ(s) (z). Mat(N,C)

If s > −1, the measure µ(s) is a probability measure, by dominated convergence theorem, the equality (35) holds. If s ≤ −1, the measure µ(s) is infinite. The radial part of µ(s) is an infinite determinantal process which corresponds to a finite-rank perturbation of determinantal probability measures as described in §5.2 in [6]. By using the asympotic formulae (31), (32) and (33) respectively in these three cases, we can check that the conditions of Proposition 3.6 in [6] are satisfied, for instance, let us check the following condition p p p p lim tr 1 − gΠ(s,n) 1 − g = tr 1 − gΠ(s) 1 − g, (36) n→∞

where Π(s,n) is the orthogonal projection onto the subspace L(s+2ns ,n−ns ) described in §5.2.1 in [6]. Combining the estimates given in Proposition 5.11 and Proposition 5.13 in [6], the integrands appeared in p p tr 1 − gΠ(s,n) 1 − g

are uniformly integrable, hence by the Heine-Mehler classical asymptotics, the equality (36) indeed holds. Now by Corollary 3.7 in [6], we have Ψg B(s,n) R
Ψg dB(s,n) →

R

Conf (0,∞)

It follows that

lim

n→∞

(37) =

R

R

Conf (0,∞)

Mat(N,C)

Mat(N,C)

R

Ψg B(s)
Ψg dB(s) .

R

f (r(n) (z))Ψg (r(n) (z))dµ(s) (z)

Mat(N,C)

Ψg (r(n) (z))dµ(s) (z)

f (r(∞) (z))Ψg (r(∞) (z))dµ(s) (z)

Mat(N,C)

Ψg (r(∞) (z))dµ(s) (z)

.

18

ALEXANDER I. BUFETOV , YANQI QIU

Moreover, by Lemma 1.14 in [6], there exists a positive bounded continuous function f such that Z Z (n) (s) lim f (r (z))dµ (z) = f (r(∞) (z))dµ(s) (z). n→∞

Mat(N,C)

Mat(N,C)

Again by Lemma 3.1, we have Z f (r(n) (z))Ψg (r(n) (z))dµ(s) (z) lim n→∞ Mat(N,C) Z (38) = f (r(∞) (z))Ψg (r(∞) (z))dµ(s) (z). Mat(N,C)



Finally, (35) follows from (37) and (38), as desired. Remark 3.3. Note that 4/x 4/x n2 x − 1 = 1 − ∼ 1 − . n2 x + 1 2n2 + 2/x 2n2

Thus under change of variable y = 4/x, in the sequel, we only consider the scaling regimes of type z x = 1 − 2. 2n 4. S CALING L IMITS OF C HRISTOFFEL -U VAROV DEFORMATIONS JACOBI O RTHOGONAL P OLYNOMIAL E NSEMBLES .

OF

In this section, we will calculate explicitly the kernels for the determinatal (s) probabilities Pg given in Proposition 3.2. For avoiding extra notation, we mention here that in the sequel, in case I the s corresponds to s − 2m in Proposition 3.2, in cases II and III, it corresponds to s + m in Propostion 3.2. For the case III, we give the result only for m = 2. Observe that in the new coordinate x = ρ(y), the kernel K(x1 , x2 ) for a locally trace class operator on L2 (R+ ) reduces to p ρ′ (y1 )ρ′ (y2 )K(ρ(y1 ), ρ(y2 )). 4.1. Explicit Kernels for Scaling Limit: Case I. Let s > −1. Consider (n) (n) a sequence ξ (n) = (ξ1 , · · · , ξm ) of m-tuples of distinct real numbers in [ξ (n) ] (−1, 1). Let ws be the weight on (−1, 1) given by (n) ws[ξ ] (t)

m m Y Y (n) (n) 2 = (ξi − t) · ws (t) = (ξi − t)2 · (1 − t)s . i=1

[s,ξ (n) ]

i=1

Let Kn (x1 , x2 ) denote the n-th Christoffel-Darboux kernel for the [ξ (n) ] weight ws . The aim of this section is to establish the scaling limit of

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

19

[s,ξ (n) ]

Kn

(x1 , x2 ) in the following regime: wi (n) ξi = 1 − 2 , 1 ≤ i ≤ m, wi > 0 are all distinct; 2n (39) zi (n) xi = 1 − 2 , zi > 0, i = 1, 2. 2n 4.1.1. Explicit formulae for orthogonal polynomials and Christoffel-Darboux [s,ξ (n) ] kernels. Let (πj )j≥0 denote the system of monic orthogonal polynomi[ξ (n) ]

als associated with the weights ws . To simplify notation, if there is no [s,ξ (n) ] (n) confusion, we denote πj by πj . (n) The monic polynomials πj ’s are given by the Christoffel formula ([15, Thm 2.5.]): (n) πj (t)

where

and

(n) Dj (t) =

= Qm

(n)

i=1 (ξi

(s) (n) Pj (ξ1 )

(n)

Dj (t)

1

, · (s) (n) − t)2 kj+2m · δj

(s) (n) Pj+1(ξ1 )

···

(s) (n) Pj+2m (ξ1 )

.. .. .. . . . (s) (n) (s) (n) (s) (n) Pj (ξm ) Pj+1(ξm ) · · · Pj+2m (ξm ) (s) (n) (s) (n) (s) (n) P˙ j (ξ1 ) P˙ j+1(ξ1 ) · · · P˙ j+2m (ξ1 )

.. .. .. . . . (s) (n) (s) (n) (s) (n) ˙ ˙ ˙ Pj (ξm ) Pj+1(ξm ) · · · Pj+2m (ξm ) (s)

Pj (t)

(s)

Pj+1(t)

···

(s)

Pj+2m (t)

;

(s) (n) (s) (n) (s) (n) Pj (ξ1 ) Pj+1(ξ1 ) · · · Pj+2m−1 (ξ1 ) .. .. .. . . . (s) (n) (s) (n) (s) (n) P (ξ ) P (ξ ) · · · P (ξ ) m j j+1 m j+2m−1 m (n) . δj (s) (n) (s) (n) (s) (n) ˙ ˙ ˙ Pj (ξ1 ) Pj+1(ξ1 ) · · · Pj+2m−1 (ξ1 ) .. .. .. . . . (s) (n) (s) (n) (s) (n) P˙ j (ξm ) P˙ j+1(ξm ) · · · P˙ j+2m−1 (ξm ) R 1 n (n) o2 [ξ(n) ] [s,ξ (n) ] Definition 4.1. Let hj = −1 πj (t) ws (t)dt. =

20

ALEXANDER I. BUFETOV , YANQI QIU

Proposition 4.2. For any j ≥ 0, we have [s,ξ (n) ] hj

(n)

(s)

=

hj

(s) (s)

kj kj+2m

·

δj+1 (n)

δj

.

Proof. By orthogonality, for any ℓ ≥ 1, we have Z 1 (s) (n) Pj+u (t)πj (t)ws (t)dt = 0. −1

Note that (n)

Dj

(n)

(s)

(s)

(s)

= δj+1 Pj (t) + linear combination of Pj+1 , · · · , Pj+2m .

Hence [s,ξ (n) ] hj

= =

1 (s) (n) kj+2mδj

1 (s) (n) kj+2mδj (n)

=

δj+1 (s)

(n)

kj+2mδj (s)

=

Z Z

(n)

(n)

Dj (t)πj (t)ws (t)dt (n)

(s)

(n)

δj+1 Pj (t)πj (t)ws (t)dt

Z n o2 1 (s) Pj (t) w (t)dt (s) s kj (n)

hj

(s) (s)

kj kj+2m

·

δj+1 (n)

δj

. 

By the Christoffel-Darboux formula (cf. [15, Thm 3.2.2]), we have: q (n) (n) n−1 (n) (n) X πj (x1 ) · πj (x2 ) (n) [ξ (n) ] (n) [ξ (n) ] (n) [s,ξ (n) ] (n) Kn (x1 , x2 ) = ws (x1 )ws (x2 ) · [s,ξ (n) ] hj j=0 q [ξ (n) ] (n) [ξ (n) ] (n) (n) (n) (n) (n) (n) (n) ws (x1 )ws (x2 ) πn(n) (x(n) 1 ) · πn−1 (x2 ) − πn (x2 ) · πn−1 (x1 ) · = . (n) (n) [s,ξ (n) ] x1 − x2 hn−1 (n)

zi 2 After change of variables xi = 1 − 2n 2 , zi ∈ [0, 4n ], i = 1, 2, and let (n) ξ takes the form as in the regime (39), these kernels can be written as:   e n[s,ξ(n) ] (z1 , z2 ) = 1 Kn[s,ξ(n) ] 1 − z1 , 1 − z2 K (40) 2n2 2n2 2n2 s (z1 z2 ) 2 · Sn (z1 , z2 ), = Qm | i=1 (z1 − wi )(z2 − wi )|

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

21

where (41)

2 2m−s−1

Sn (z1 , z2 ) = (2n )

(n) n−1 X Dj (1 −

(n) z1 )Dj (1 2n2

(s) (s)

−

z2 ) 2n2

hj kj+2m (n) (n) δj δj+1 (s) k

j=0

,

j

or equivalently (42) Sn (z1 , z2 ) =

(2n2 )2m−s h i2 × (s) (s) hn−1 kn+2m (n) δn (s) kn−1

×

(n) Dn (1

−

z1 ) 2n2

(n)

· Dn−1 (1 −

z2 ) 2n2

(n)

− Dn (1 − z2 − z1

z2 ) 2n2

(n)

· Dn−1 (1 −

z1 ) 2n2

.

4.1.2. Scaling limits. To obtain the scaling limit of the Christoffel-Darboux e n[s,ξ(n) ] (z1 , z2 ), we shall investigate the asymptotics of the formulae kernels K (41) or (42). These two representations (41) and (42) will yield different representations of the scaling limit: an integrable form and an integral form. The following formulae are well-known ([15, p.63, p.68]): 1 Γ(2j + s + 1) 2s+1 (s) , h = . j 2j · j! Γ(j + s + 1) 2j + s + 1 The following lemma will be used frequently in the sequel. (s)

kj =

(43)

Lemma 4.3. Let p ∈ Z, then

(s)

lim

kκn +p

n→∞

(s) kκn

= 2p .

Proof. It is an easy consequence of (43) and the Stirling’s approximation formula for Gamma functions, here we can also use the following convenient formula: Γ(n + a) for all a ∈ R, lim a = 1. n→∞ n Γ(n)  Proposition 4.4. If the sequence ξ (n) satisfies (39) , then 22ms 2 (s,w) = lim n2m −2sm−3m δκ(n) CI (κ), n 1+s n→∞ (w1 · · · wm ) where (s,w) C (κ) = W (Js,w1 , · · · , Js,wm , Jes+1,w1 , · · · Jes+1,wm )(κ) I

and

√ Js,wi (κ) := Js (κ wi ),

√ Jes+1,wi (κ) = Jes+1 (κ wi ).

22

ALEXANDER I. BUFETOV , YANQI QIU

Moreover, the convergence is uniform as long as κ is in a compact subset of (0, ∞). (s; ℓ)

[ℓ]

Proof. To simplify notation, we denote ∆P,n by ∆P,n in this proof. By the multi-linearity of the determinant on columns, we have [0] (n) [1] (n) [2m−1] (n) ∆ P,κn (ξ1 ) ∆P,κn (ξ1 ) · · · ∆P,κn (ξ1 ) . . . .. .. .. ∆[0] (ξ (n) ) ∆[1] (ξ (n) ) · · · ∆[2m−1] (ξ (n) ) m m m P,κ P,κ P,κ n n n . = δκ(n) n [0] (n) [1] (n) [2m−1] (n) ˙ ˙ ˙ ∆P,κ (ξ1 ) ∆P,κ (ξ1 ) · · · ∆P,κ (ξ1 ) n n n .. .. .. . . . [0] (n) [1] (n) [2m−1] (n) ∆ ˙ ˙ ˙ P,κn (ξm ) ∆P,κn (ξm ) · · · ∆P,κn (ξm ) Multiplying the matrix used in the above formula on right by the diagonal matrix diag(n−s , n1−s , · · · , n2m−1−s ) and on left by the diagonal matrix diag(1, · · · , 1, n−2 , · · · , n−2 ) and taking determinant, we obtain that | {z } | {z }

m terms 2m2 −2sm−3m (n) n δκn



m terms

equals to the following determinant

[2m−1] (n) · · · n2m−1−s ∆P,κn (ξ1 ) .. .. .. . . . [0] (n) [1] (n) [2m−1] (n) n−s ∆P,κn (ξm ) n1−s ∆P,κn (ξm ) · · · n2m−1−s ∆P,κn (ξm ) . (n) (n) 2m−3−s ˙ [2m−1] (n) −1−s ˙ [1] −2−s ˙ [0] ∆P,κn (ξ1 ) ∆P,κn (ξ1 ) · · · n n ∆P,κn (ξ1 ) n .. .. .. . . . (n) (n) −2−s ˙ [0] −1−s ˙ [1] 2m−3−s ˙ [2m−1] (n) n ∆P,κn (ξm ) n ∆P,κn (ξm ) · · · n ∆P,κn (ξm ) [0] (n) n−s ∆P,κn (ξ1 )

[1]

(n)

n1−s ∆P,κn (ξ1 )

Applying Propositions 2.2 and 2.3, we obtain the desired formula. The last statement follows from the uniform convergences in Propositions 2.2, 2.3.  (n)

Proposition 4.5. If the sequences xi 2 −m−2ms−s

lim n2m

n→∞

where (s,w)

AI

(n)

Dκ(n) (xi ) = n

and ξ (n) satisfy (39) , then

22ms+s − 2s (s,w) z · AI (κ, zi ), i 1+s (w1 · · · wm )

  (κ, zi ) = W Js,w1 , · · · , Js,wm , Jes+1,w1 , · · · , Jes+1,wm , Js,zi (κ).

Moreover, the convergence is uniform as long as κ is in a compact subset of (0, ∞).

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

23



Proof. The proof is similar to that of Proposition 4.4. (n)

Definition 4.6. Define the column vector function θ j (t) by  T (n) (s) (n) (s) (n) (s) (n) (s) (n) (s) ˙ ˙ θ j (t) = Pj (ξ1 ), · · · , Pj (ξm ), Pj (ξ1 ), · · · , Pj (ξm ), Pj (t) . (n)

Proposition 4.7. If the sequences xi and ξ (n) satisfy (39), then 2 (n) (n) (n) (n) (n) (n) (n) (x ) · · · θ (x ) θ (x ) − θ (x ) lim n1+2m −m−2ms−s θ κ(n) κn +2m−1 i κn +2m i κn −1 i i n n→∞

22ms+s − 2s (s,w) · BI (κ, zi ), z (w1 · · · wm )1+s i (s,w) (2m−1) (2m+1) ′ where BI (κ, zi ) = η s,zi (κ) η s,zi (κ) · · · η s,zi (κ) η s,zi (κ) and η s,zi (κ) is the column vector  √ √ √ √ √ T Js (κ w1 ), · · · , Js (κ wm ), Jes+1(κ w1 ), · · · , Jes+1(κ w1 ), Js (κ zi ) . =

Proof. The proof is similar to that of Proposition 4.4, we emphasize that in the proof we used the elementary fact [2m+1]

(s)

(s)

∆P,κn −1 =Pκn +2m + (−1)2m+1 Pκn −1

(s)

(s)

+ linear combination of Pκ(s) , Pκn+1 , · · · , Pκn +2m−1 . n

Remark 4.8. By the property of determinant, it is easy to see that ∂ (s,w) (s,w) AI (κ, zi ) = BI (κ, zi ). ∂κ Theorem 4.9. In the regime (39) , we have e [s,ξ(n) ] (z1 , z2 ) lim K n

n→∞ (s,w) (s,w) (s,w) (s,w) AI (1, z1 )BI (1, z2 ) − AI (1, z2 )BI (1, z1 ) = Q .  (s,w) 2 (z − w )(z − w ) · C 2 m (1) · (z − z ) 1 i 2 i 1 2 I i=1 [s,ξ]

We denote this kernel by K∞ (z1 , z2 ).

Proof. It is easy to see that (n) (n) (n) (n) (n) (n) (n) (x ) · · · θ (x ) θ (x ) Dn−1 (xi ) = θ (n) , n n+2m−1 i n−1 i i (n)

(n)

(n)

(n)

hence Dn (xi ) − Dn−1 (xi ) equals to (n) (n) (n) (n) (n) (n) (n) (n) θ n (xi ) · · · θ n+2m−1 (xi ) θ n+2m (xi ) − θ n−1 (xi ) .



24

ALEXANDER I. BUFETOV , YANQI QIU

Note that (n)

(n)

(n)

(n)

(n)

(n)

Dn(n) (x1 )Dn−1 (x2 ) − Dn(n) (x2 )Dn−1 (x1 ) h i (n) (n) (n) (n) =Dn(n) (x2 ) Dn(n) (x1 ) − Dn−1 (x1 ) h i (n) (n) (n) (n) − Dn(n) (x1 ) Dn(n) (x2 ) − Dn−1 (x2 ) .

Now applying Propositions 4.5 and 4.7, we obtain that h i (n) (n) (n) (n) (n) (n) (n) (n) 1+4m2 −2m−4ms−2s Dn (x1 )Dn−1 (x2 ) − Dn (x2 )Dn−1 (x1 ) lim n n→∞ s  24ms+2s (z1 z2 )− 2  (s,w) (s,w) (s,w) (s,w) = AI (1, z2 )BI (1, z1 ) − AI (1, z1 )BI (1, z2 ) . (w1 · · · wm )2+2s

Combining with Proposition 4.4, we deduce that lim Sn (z1 , z2 )

n→∞

=(z1 z2 )

− 2s

(s,w)

·

AI

(s,w)

(1, z1 )BI

(s,w)

(s,w)

(1, z2 ) − AI (1, z2 )BI  (s,w) 2 2 CI (1) (z1 − z2 )

(1, z1 )



Substituting the above formula in (40), we get the desired result. (s,ξ)

Theorem 4.10. The kernel K∞

(z1 , z2 ) has the following integral form:

K∞(s,ξ) (z1 , z2 ) 1

.

= Q 2 m (z − w )(z − w ) i 2 i i=1 1

Z

1

(s,w)

AI

0

(s,w)

(t, z1 )AI (t, z2 ) tdt.  (s,w) 2 CI (t)

Proof. Let us fix z1 , z2 > 0. For any ε > 0, we can divide the sum in (41) into two parts: 2 2m−s−1

Sn (z1 , z2 ) = (2n )

{z

|

⌊nε⌋−1

X j=0

2 2m−s−1

· · · + (2n ) }

=: In (ε)

|

(n)

 (2n )

2 2m−s−1

⌊nε⌋ ,1 n

|

(n)

j=⌊nε⌋

=: IIn (ε)

The second term IIn (ε) can be written as an integral: Z IIn (ε) = h

{z

n−1 X

(n)

(s)

h⌊nt⌋ k⌊nt⌋+2m (n) (n) δ⌊nt⌋ δ⌊nt⌋+1 (s) k ⌊nt⌋

{z

=: Tn (t)

}

(n)

D⌊nt⌋ (x1 )D⌊nt⌋ (x2 ) (s)

···.

·n }

dt.

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

25

By Propositions 4.4 and 4.5, we have the uniform convergence for t ∈ [ε, 1]: s

(z1 z2 )− 2 (s,w) (s,w) lim Tn (t) = A (t, z1 )AI (t, z2 )t, n→∞ 2(C [s,w](t))2 I

hence as n → ∞, IIn (ε) tends to Z 1 s (z1 z2 )− 2 (s,w) (s,w) II∞ (ε) = AI (t, z1 )AI (t, z2 )tdt. (s,w) 2 ε 2(CI (t)) For the first term In (ε), we use Christoffel-Darboux formula to write it as (n)

(2n2 )2m−s (s)

(s)

h⌊nε⌋−1 k⌊nε⌋+2m (s)

k⌊nε⌋−1

(n)

(n)

(n)

(n)

(n)

(n)

(n)

D⌊nε⌋ (x1 ) · D⌊nε⌋−1 (x2 ) − D⌊nε⌋ (x2 ) · D⌊nε⌋−1 (x1 ) .  (n) 2 δ⌊nε⌋ (z2 − z1 )

By similar arguments as in the proof of Theorem 4.9, lim In (ε) = I∞ (ε),

n→∞

where I∞ (ε) is given by the formula I∞ (ε)

(s,w)

s

(z1 z2 )− 2 AI = · 2

(s,w)

(ε, z1 )BI

Hence for any ε > 0, we have (n)

(s,w)

(s,w)

(ε, z2 ) − AI (ε, z2 )BI  (s,w) 2 CI (ε) (z1 − z2 )

(ε, z1 )

· ε.

(n)

lim Sn (z1 , z2 ) = I∞ (ε) + II∞ (ε).

n→∞

The theorem is completely proved if we can establish limε→0 I∞ (ε) = 0. This is given by the following lemma.  Lemma 4.11. For any z1 , z2 > 0, we have (s,w)

lim

ε→0+

AI

(s,w)

(ε, z1 )BI

(s,w)

(s,w)

(ε, z2 ) − AI (ε, z2 )BI  (s,w) 2 CI (ε)

(ε, z1 )

· ε = 0.

Proof. To simplify notation, let us denote Fi = Js,wi and Gi = Jes+1,wi . We have [s,w] CI (ε) = W (F1 , · · · , Fm , G1 , · · · , Gm )(ε). By (12), we have Gi (ε) = −εFi′ (ε) + sFi (ε). If we denote Hi (ε) = −εFi′ (ε), then C [s,w](ε) = W (F1 , · · · , Fm , H1 , · · · , Hm )(ε).

26

ALEXANDER I. BUFETOV , YANQI QIU

We can write Fi (ε) = aν(i)

P∞

(i)

aν ν=0 ν!

ε2ν+s and Hi (ε) =

√ (−1)ν ( wi )2ν+s , = 2ν+s 2 Γ(ν + s + 1)

P∞

(i)

bν ν=0 ν!

ε2ν+s , with

(i) b(i) ν = −(2ν + s)aν .

Define entire functions: fi (x) =

∞ (i) X aν ν=0

ν!

ν

x ,

hi (x) =

∞ (i) X bν ν=0

ν!

xν .

Then Fi (ε) = εs fi (ε2 ) and Hi (ε) = εs hi (ε2 ). Using the identity W (gf1, · · · , gfn )(x) = g(x)n · W (f1 , · · · , fn )(x), we obtain that (s,w)

CI

  (ε) = ε2ms · W f1 (x2 ), · · · , fm (x2 ), h1 (x2 ), · · · , hm (x2 ) (ε).

An application of the following identity

⌊n/2⌋ i X (2x)n−2k dn h 2 f (x ) = n! f (n−k) (x2 ) dxn k!(n − 2k)! k=0

yields (s,w)

CI

(ε) = 2m(2m−1) ε2ms+m(2m−1) W (f1 , · · · , fm , h1 , · · · , hm )(ε2 ).

We state the following simple auxiliary Lemma 4.12. W (f1 , · · · , fm , h1 , · · · , hm )(z) is an entire function and does not vanish at z = 0. Before proving Lemma 4.12, we derive from it Lemma 4.11. Indeed, from Lemma 4.12 we have (s,w)

CI

(ε) ≍ ε2ms+m(2m−1) as ε → 0.

Similarly, (s,w)

AI (ε, zi ) ≍ ε(2m+1)s+m(2m+1) as ε → 0. By Remark 4.8, we also have (s,w)

BI

(ε, zi ) ≍ ε(2m+1)s+m(2m+1)−1 as ε → 0.

Hence as ε → 0, we have A(s,w)(ε, z )B (s,w) (ε, z ) − A(s,w) (ε, z )B (s,w) (ε, z ) I 1 2 2 1 I I I · ε . ε4m+2s . h i2 (s,w) CI (ε) Since we always have 4m + 2s > 0, Lemma 4.11 is proved.

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

27

Now we turn to the proof of the Lemma 4.12. By definition, we know that fi , hi are entire functions, hence W (f1 , · · · , fm , h1 , · · · , hm ) is also entire. It is easily to see that W (f1 , · · · , fm , h1 , · · · , hm )(0) equals to (1) (1) (1) (1) a0 a a · · · a 1 2 2m−1 .. .. .. .. . . . . (m) (m) (m) (m) a0 a a · · · a 1 2 2m−1 , b(1) b(1) b(1) · · · b(1) 0 1 2 2m−1 . .. .. .. .. . . . b(m) b(m) b(m) · · · b(m) 0 1 2 2m−1

which is in turn given by a non-zero multiple of det W , where   1 w1 w12 · · · w12m−1 .. .. ..   .. . . .   .   1 w 2 2m−1 w · · · w   m m m . W =    0 1 2w1 · · · (2m − 1)w12m−2    . .. .. ..   .. . . . 2m−2 0 1 2wm · · · (2m − 1)wm

We claim that det W 6= 0. Indeed, let θ = (θ0 , θ1 , · · · , θ2m−1 )T be such that W θ = 0. In other words, we have 2m−1 X

θk wik

= 0,

2m−1 X k=0

k=0

kθk wik−1 = 0, for 1 ≤ i ≤ m.

P2m−1 k Let Θ be the polynomial given by Θ(x) = k=0 θk x , then the above equations imply that w1 , · · · , wm are distinct roots of Θ, each wi has multiplicity at least 2. Since deg Θ ≤ 2m − 1, we must have Θ ≡ 0 and hence θ = 0. This shows that W is invertible, hence has a non-zero determinant.  4.2. Explicit Kernels for Scaling Limit: Case II. Consider a sequence of (n) (n) m-tuples of distinct positive real numbers r (n) = (r1 , · · · , rm ) and the (r (n) ) modified weights ws given as follows: ws(r

(n) )

(t) = Qm

ws (t) (n)

i=1 (1 + ri

− t)

= Qm

(1 − t)s

(n)

i=1 (1 + ri

− t)

(r (n) )

The n-th Christoffel-Darboux kernel associated with ws (n) Πn (x1 , x2 ).

.

is denoted by

28

ALEXANDER I. BUFETOV , YANQI QIU (n)

(n)

(n)

We will investigate the scaling limit of Πn (x1 , x2 ) in the regime: zi , zi > 0, i = 1, 2. 2n2 vi = 2 , 1 ≤ i ≤ m, and vi > 0 are all distinct. 2n (n)

xi

(44)

(n)

ri

=1−

4.2.1. Explicit formulae for orthogonal polynomials and Christoffel-Darboux kernels. The Christoffel-Uvarov formula implies that the following polyno(n) (r (n) ) mials qj for j ≥ m are orthogonal with respect to ws : (s) (n) Qj−m (1 + r1(n) ) · · · Q(s) j (1 + r1 ) .. .. (n) . qj (t) = (s) . (n) (s) (n) Qj−m (1 + rm ) · · · Qj (1 + rm ) (s) (s) ··· Pj (t) Pj−m (t) (n)

.

For 0 ≤ j < m, we also denote by qj the j-th monic orthogonal polynomial, here we will not give its explicit formula. Denote

(n)

dj

(s) (n) (s) (n) Q j−m (1 + r1 ) · · · Qj−1 (1 + r1 ) .. .. = . . (s) (n) (s) (n) Q j−m (1 + rm ) · · · Qj−1 (1 + rm )

(s,r (n) )

Denote by kj (s,r (n) )

by kj

(n)

.

the leading coefficient of qj . When j ≥ m, it is given

(n) (s)

= d j kj .

(s,r (n) )

Definition 4.13. Define hj

=

R 1 n (n) o2 (r(n) ) qj (t) ws (t)dt. −1

Proposition 4.14. For any j ≥ m, we have (n) (n)

(s,r (n) ) hj

=

(s) (s)

dj dj+1kj hj−m (s)

kj−m

.

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

29

Proof. Let j ≥ m, by the orthogonality, we have (s,r (n) )

=

hj Z

(n) (n) (n) (s) qj (t)dj Pj (t)ws(r ) (t)dt

Z

(n) (s) =dj kj

(n) qj (t)(−1)m

(n) (s) =(−1)m dj kj (n) (s) =(−1)m dj kj (n) (n)

=

(s)

kj−m (n) (n)

=

(s)

dj dj+1kj

(n)

(1 + ri

i=1

Z

Z

m Y

=

(n) (s) d j kj

(n)

Z

(n)

qj (t)tj ws(r

− t)tj−m ws(r

(n) )

(n) )

(t)dt

(t)dt

qj (t)tj−m ws (t)dt (n)

(s)

(−1)m+2 dj+1 Pj−m(t)tj−m ws (t)dt

Z n o2 (s) Pj−m (t) ws (t)dt

(s) (s)

dj dj+1kj hj−m (s)

kj−m

.  (n)

zi (n) be as in the regime By change of variables, xi = 1 − 2n 2 , and let r (44), the Christoffel-Darboux kernels are given by the formula s

(45) where (46)

e (n) Π n (z1 , z2 )

= Qm

(z1 z2 ) 2 1

Σn (z1 , z2 ),

1

2 2 i=1 (vi + z1 ) (vi + z2 )

2 m−s−1

Σn (z1 , z2 ) = (2n )

n−1 (n) X qj (1 −

(n) z1 )q (1 2n2 j

(n) (n)

(s) (s)

−

z2 ) 2n2

dj dj+1 kj hj−m

j=0

,

(s)

kj−m

or equivalently Σn (z1 , z2 ) (n)

(47)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

qn (x1 )qn−1 (x2 ) − qn (x2 )qn−1 (x1 ) (2n2 )m−s = . · (s) (s)  z2 − z1 (n) 2 hn−1−m kn dn (s) kn−1−m

4.2.2. Scaling limits. Now we investigate the scaling limits. Proposition 4.15. In the regime (44) we have lim n

n→∞

m(m−1) −ms 2

s

(s,v)

ms −2 d(n) CII (κ), κn = 2 (v1 · · · vm )

30

ALEXANDER I. BUFETOV , YANQI QIU

lim n

n→∞

m(m+1) −(m+1)s 2

s

(n)

−s

(s,v)

qκ(n) (xi ) = 2(m+1)s (v1 · · · vm )− 2 zi 2 AII (κ, zi ), n

where

  (s,v) CII (κ) = W Ks,v1 , · · · , Ks,vm (κ),   (s,v) AII (κ, z) = W Ks,v1 , · · · , Ks,vm , Js,z (κ), √ √ and Ks,vi (κ) = Ks (κ vi ), Js,z (κ) = Js (κ z).

Proof. For ℓ ≥ 1, we have (s,ℓ)

(s)

(s)

(s)

∆Q,n = Qn+ℓ + (−1)ℓ Q(s) n + linear combination of Qn+1 , · · · , Qn+ℓ−1 . (s,ℓ)

The same is true for ∆P,n and with the same coefficients. Hence for kn ≥ m, we have (s,0) (n) (s,m−1) (n) ∆ Q,κn −m (1 + r1 ) · · · ∆Q,κn −m (1 + r1 ) .. .. = dκ(n) ; . . n (s,0) (n) (s,m−1) (n) ∆ Q,κn −m (1 + rm ) · · · ∆Q,κn −m (1 + rm ) ∆(s,0) (1 + r (n) ) · · · ∆(s,m) (1 + r (n) ) Q,κn−m 1 1 Q,κn −m .. .. . . (n) (x ) = qκ(n) (s,0) (n) (s,m) (n) i n ∆Q,κn−m (1 + rm ) · · · ∆Q,κn −m (1 + rm ) (n) (s,m) (n) ∆(s,0) ··· ∆P,κn−m (xi ) P,κn −m (xi )

.

The proposition is completely proved by applying the same arguments as in the proof of Proposition 4.4 and by applying Propositions 2.2, 2.3, 2.4 and 2.5.  Proposition 4.16. In the regime (44), we have h i m(m+1) (n) (n) (n) lim n 2 −(m+1)s+1 qκ(n) (x ) − q (x ) κn −1 i i n n→∞

s

−s

(s,v)

= 2(m+1)s (v1 · · · vm )− 2 zi 2 BII (κ, zi ),

where ∂ (s,v) (s,v) A (κ, z) BII (κ, z) = ∂κ II (m+1) (m−1) (κ) , (κ), φs,z = φs,z (κ), φ′s,z (κ), · · · , φs,z  √ T √ √ and φs,z (κ) is the column vector Ks (κ v1 ), · · · Ks (κ vm ), Js (κ z) .

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

31

Proof. To simplify notation, we show the proposition in the case κn = n, the proof in the general case is similar. Define column vector  T (n) (s) (n) (s) (s) (n) βj (t) = Qj (1 + r1 ), · · · , Qj (1 + rm ), Pj (t) . Then for i = 1, 2, (n) (n) (n) (n) (n) (n) (n) qn(n) (xi ) = βn−m (xi ) · · · βn−1 (xi ) βn (xi ) ; (n) (n) (n) (n) (n) (n) (n) (n) qn−1 (xi ) = βn−1−m (xi ) · · · βn−2 (xi ) βn−1 (xi ) (n) (n) (n) (n) (n) (n) =(−1)m βn−m (xi ) · · · βn−1 (xi ) βn−1−m (xi ) .

Hence

(n)

(n)

(n)

qn(n) (xi ) − qn−1 (xi (n) (n) = βn−m (xi ) · · · ∆(s,0) (1 + r (n) ) Q,n−m 1 . .. = (s,0) (n) ∆Q,n−m (1 + rm ) (n) ∆(s,0) P,n−m (xi )

)

(n) (n) βn−1 (xi )

(n) (n) βn (xi )

+

(n) (n) (−1)m+1 βn−1−m (xi )



(s,m−1) (n) (s,m+1) (n) · · · ∆Q,n−m (1 + r1 ) ∆Q,n−1−m (1 + r1 ) .. .. . . (s,m−1) (n) (s,m+1) (n) . · · · ∆Q,n−m (1 + rm ) ∆Q,n−1−m (1 + rm ) (s;m−1) (n) (s;m+1) (n) ··· ∆P,n−m (xi ) ∆P,n−1−m (xi )



We finish the proof by using Propositions 2.2, 2.3, 2.4 and 2.5. Combining Propositions 4.15 and 4.16, we obtain

Corollary 4.17. In the regime (44), we have i h (n) (n) (n) (n) (n) (n) (n) (x )q (x ) (x )q (x ) − q lim nm(m+1)−2(m+1)s+1 qκ(n) 2 κn −1 1 1 κn −1 2 κn n n→∞ (s,v) A (κ, z1 ) −B (s,v) (κ, z1 ) s s − − II II = 22(m+1)s (v1 · · · vm )−s z1 2 z2 2 (s,v) . (s,v) A (κ, z ) −B (κ, z ) II

(n)

(n)

(n)

(n)

2

(n)

2

II

(n)

(n)

(n)

Proof. We first write qκn (x1 )qκn −1 (x2 ) − qκn (x2 )qκn −1 (x1 ) as q (n) (x(n) ) q (n) (x(n) ) q (n) (x(n) ) q (n) (x(n) ) − q (n) (x(n) ) κn κn 1 κn 1 1 κn −1 1 κn −1 1 = (n) (n) (n) (n) (n) (n) (n) (n) (n) qκn (x2 ) qκ(n) (x2 ) qκn (x2 ) qκn −1 (x2 ) − qκn (x2 ) n −1 The corollary now follows from Propositions 4.15 and 4.16.

.



32

ALEXANDER I. BUFETOV , YANQI QIU

Theorem 4.18. In the regime (44), we obtain the scaling limit e (n) (z1 , z2 ) Π(s,v) (z1 , z2 ) := lim Π ∞

n n→∞ (s,v) (s,v) (s,v) (s,v) AII (1, z1 )BII (1, z2 ) − AII (1, z2 )BII (1, z1 ) = Q . p  (s,v) 2 2 m (v + z )(v + z ) · C (1) · (z − z ) i 1 i 2 1 2 II i=1

Proof. By (47) and Proposition 4.15, Corollary 4.17, we have

lim Σn (z1 , z2 ) n o (s,v) (s,v) (s,v) (s,v) − 2s AII (1, z1 )BII (1, z2 ) − AII (1, z2 )BII (1, z1 ) (z1 z2 ) . =  (s,v) 2 2 CII (1) (z1 − z2 ) n→∞

Combining this with (45), we get the desired result.



(s,v)

Proposition 4.19. Let s > m − 1, s ∈ / N. The kernel Π∞ (z1 , z2 ) has the following integral form: Π(s,v) ∞ (z1 , z2 ) 1

= Qm p 2 i=1 (vi + z1 )(vi + z2 )

Z

0

1

(s,v)

(s,v)

AII (κ, z1 ) · AII (κ, z2 ) κdκ.  (s,v) 2 CII (κ)

Proof. The proof is similar to that of Theorem 4.10, a slight difference is, instead of using Lemma 4.11, we shall use the following Lemma 4.20.  Lemma 4.20. Let s > m − 1, s ∈ / N. For any z1 , z2 > 0, we have (s,v)

(s,v)

(s,v)

(s,v)

A (ε, z1 )BII (ε, z2 ) − AII (ε, z2 )BII (ε, z1 ) · ε = 0. lim+ II  (s,v) 2 ε→0 CII (ε)   (s,v) Proof. Recall that CII (ε) = W Ks,v1 , · · · , Ks,vm (ε) (ε). By McDonald definition of Ks (z), namely ∞ X ( 21 z)2ν+s π I−s (z) − Is (z) , Is (z) = , Ks (z) = 2 sin sπ ν!Γ(ν + s + 1) ν=0

we can write

∞ ∞ (i) (i) X X αν 2ν βν 2ν  π −s 2s , Ks,vi (ε) = Ks (ε vi ) = ε ε −ε ε 2 sin(sπ) ν! ν! ν=0 ν=0 {z } |

√

:=Ai (ε2 )

where

αν(i)

√ ( 12 vi )2ν−s = , Γ(ν − s + 1)

βν(i)

√ ( 21 vi )2ν+s = . Γ(ν + s + 1)

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

Thus

33

  (s,v) CII (ε) = Consts × ε−ms W A1 (x2 ), · · · , Am (x2 ) (ε)   m(m−1) −ms+ 2 = Consts × ε W A1 , · · · , Am (ε2 ).

By the assumption s > m − 1, we know that Ai are all differentiable up to order at least m − 1 on the neighbourhood of 0, hence W (A1 , · · · , Am )(ε) is continuous, and 1 v1 · · · v1m−1   . . .. 6= 0. W A1 , · · · , Am (0) = non-zero term × .. .. . 1 v · · · v m−1 m m Hence

(s,v)

CII (ε) ≍ ε−ms+ In a similar way, we can show that (s,v)

AII (ε, zi ) ≍ ε−(1+m)s+

m(m−1) 2

, as ε → 0.

m(m+1) +2[s−(m−1)] 2

(s,v)

BII (ε, zi ) ≍ ε−(1+m)s+

m(m+1) +2(s−m) 2

, as ε → 0.

, as ε → 0.

Hence A(s,v) (ε, z )B (s,v) (ε, z ) − A(s,v) (ε, z )B (s,v) (ε, z ) II 1 2 2 1 II II II ε . ε2s−2m+3 .  (s,v) 2 C (ε) II

Since 2s − 2m + 3 > 0, the lemma is completely proved.



Remark. Let us consider the case where −1 < s < 0 and m = 1. Let us denote I (s,v) (κ, z1 , z2 ) Ks (κ√v) √vK ′ (κ√v) Ks (κ√v) √vK ′ (κ√v) s s (48) Js (κ√z1 ) √z1 Js′ (κ√z1 ) Js (κ√z2 ) √z2 Js′ (κ√z2 ) = · κ. p  √ 2 2 (v + z1 )(v + z2 ) · Ks (κ v) For any ε, we divide the following sum into two parts: n−1 (n) (n) q 1 X qj (x1 )qj (x2 ) (r (n) ) (n) (r (n) ) (n) (n) Πn (z1 , z2 ) = 2 w (x )w (x2 ) s s 1 (n) (s,r ) 2n j=0 hj ⌊nε⌋−1 n−1 1 X 1 X = 2 ···. ···+ 2 2n j=0 2n j=⌊nε⌋ {z } | | {z } (1)

:=Sn (ε,z1 ,z2 )

(2)

:=Sn (ε,z1 ,z2 )

34

ALEXANDER I. BUFETOV , YANQI QIU

From the previous propositions and Theorem 4.18, we know that the following limits all exist lim Sn(1) (ε, z1 , z2 ),

lim Sn(2) (ε, z1 , z2 ),

n→∞

lim Π(n) n (z1 , z2 ).

n→∞

n→∞

By denoting (1) (2) S∞ (ε, z1 , z2 ) = lim Sn(1) (ε, z1 , z2 ), S∞ (ε, z1 , z2 ) = lim Sn(2) (ε, z1 , z2 ), n→∞

n→∞

we have for any ε > 0, (49)

(1) (2) Π(s,v) ∞ (z1 , z2 ) = S∞ (ε, z1 , z2 ) + S∞ (ε, z1 , z2 ).

If z1 = z2 , then every term is positive, hence Z 1 (s,v) (2) Π∞ (z1 , z1 ) ≥ S∞ (ε, z1 , z1 ) = I (s,v) (κ, z1 , z1 )dκ. ε

By Cauchy-Schwarz inequality, we can show that

|I (s,v) (κ, z1 , z2 )|2 ≤ I (s,v) (κ, z1 , z1 ) · I (s,v) (κ, z2 , z2 ).

Again by Cauchy-Schwarz inequality, we see that κ → I (s,v) (κ, z1 , z2 ) is integrable on (0, 1). Combining this fact with (49), we see that the (1) limit limε→0+ S∞ (ε, z1 , z2 ) always exists. Let us denote this limit by (1) S∞ (0, z1 , z2 ). (1) Now we show that S∞ (0, z1 , z2 ) is not identically zero. Let z1 = z2 , then for any ε > 0, we have  (n) 2 q (x 1 (n) 0 (n) 1 ) ws(r ) (x1 ) Sn(1) (ε, z1 , z1 ) ≥ 2 (n) ) (s,r 2n h 0

(n)

1 (1 − x1 )s 1 = 2 R 1 (r(n) ) (n) 2n w (t)dt 1 + r (n) − x1 −1 s zs 1 = 1 . R 1 (r(n) ) v + z1 (2n2 )s ws (x)dx −1

We have

2 s

(2n )

n→∞

−−−→ Hence

Z

1

(n) ws(r ) (x)dx

Z−1∞ 0

=

Z

0

4n2

ts dt v+t

ts dt = v s Γ(−s)Γ(s + 1). v+t

(1) S∞ (0, z1 , z1 ) ≥

1 z1s 6= 0. · v s Γ(−s)Γ(s + 1) v + z1

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

35

Definition 4.21. For −1 < s < 0, define a positive function on R∗+ : r zs 1 (s,v) . N (z) := s (v Γ(−s)Γ(s + 1))1/2 v + z Proposition 4.22. For m = 1 and −1 < s < 0, we have Z 1 (s,v) (s,v) (s,v) Π∞ (z1 , z2 ) = N (z1 )N (z2 ) + I (s,v) (κ, z1 , z2 )dκ. 0

Proof. By (49), it suffices to show that (1) S∞ (0, z1 , z2 ) = N

(s,v)

(z1 )N

(s,v)

(z2 ). (1)

By similar arguments in the proof of Theorem 4.18, S∞ (ε, z1 , z2 ) is given by the formula (50) (s,v)

(1) S∞ (ε, z1 , z2 )

(s,v)

(s,v)

(s,v)

A (ε, z1 )BII (ε, z2 ) − AII (ε, z2 )BII (ε, z1 ) . = ε · II p  (s,v) 2 2 (v + z1 )(v + z2 ) CII (ε) (z1 − z2 )

For m = 1, we have

√ ′ √ √ Ks (ε v) vKs (ε v) (s,v) AII (ε, zi ) = , Js (ε√zi ) √zi J ′ (ε√zi ) s √ (s,v) (s,v) (s,v) ∂ and BII (ε, zi ) = ∂ε AII (ε, zi ), CII (ε) = Ks (ε v). By the differential ′ ′ g′ 1 g g formula ( fg )′ = f g−f , we have = 2 2 g g f f′  √  √ 2 ∂ Js (ε zi ) (s,v) √ AII (ε, zi ) = [Ks (ε v)] , ∂ε Ks (ε v) and (s,v)

(s,v)

(s,v)

(s,v)

AII (ε, z1 )BII (ε, z2 ) − AII (ε, z2 )BII (ε, z1 ) ! (s,v) A ∂ (ε, z ) 2 (s,v) II =[AII (ε, z1 )]2 . ∂ε A(s,v) II (ε, z1 ) Hence (s,v)

(51)

(s,v)

(s,v)

(s,v)

AII (ε, z1 )BII (ε, z2 ) − AII (ε, z2 )BII (ε, z1 )  √ 2 Ks (ε v)   √  Js (ε z2 ) ∂   √ 2 √
2 ∂ Js (ε z1 ) ∂  ∂ε Ks (ε√v)   √  . √ = Ks (ε v) ∂ε Ks (ε v) ∂ε  ∂ Js (ε √z1 )  ∂ε

Ks (ε v)

36

ALEXANDER I. BUFETOV , YANQI QIU

As ε → 0+, we have √
2 Ks (ε v) ∼

(52)

(53)



∂ ∂ε



√

( 2v )s π 2 sin(sπ) Γ(s + 1)

√ 2 Js (ε z1 ) √ ∼ Ks (ε v)

!2

ε2s ,

!2 √ z1 s ) 2√ π Γ(−s)( 2v )3s 2 sin(sπ) 2Γ(s + 1)(

ε−4s−2 ,

  J (ε√z )   s 2 ∂ √ 2s  √ s z2 ∂  ∂ε Ks (ε√v)  v Γ(−s) z1 − z2 2s+1  √  ∼ √ (54) ε . ∂ε  ∂ Js (ε √z1 )  z1 Γ(s + 1) 2 2 ∂ε

Ks (ε v)

For example, let us check the asymptotic formula (54). We have  √ s √ Js (ε zi ) zi F (ε2 zi ) 2 sin(sπ) √ = , Ks (ε v) π 2 G (ε2 ) − ε−2s H (ε2 )

where F , G , H are entire functions given by F (z) =

∞ X ν=0

(−1)ν ( 4z )ν , ν!Γ(ν + s + 1)

 √ s X ∞ ( z4 )ν v G (z) = , 2 ν!Γ(ν + s + 1) ν=0

√ −s X ∞ ( 4z )ν v H (z) = . 2 ν!Γ(ν − s + 1) ν=0 It follows that    √    Js (ε z2 ) F (xz2 ) ∂ ∂  √ s √   z ∂ε Ks (ε v) ∂  ∂x G (x)−x−s H (x)  2 ∂  √  = √ 2 2ε ·   (ε ) . F (xz1 ) ∂ε  ∂ Js (ε √z1 )  z1 ∂x ∂ −s ∂ε Ks (ε v) ∂x G (x)−x H (x) {z } | =: Q(ε2 )

For i = 1, 2, let us denote

  s+1 Qi (x) =zi F (xzi ) x G (x) − xH (x)   − F (xzi ) xs+1 G ′ (x) + sH (x) − xH ′ (x) , ′

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

37

h i Q2 (x) ∂ . Note that Q1 (0) = Q2 (0) = −sF (0)H (0). Now then Q(x) = ∂x Q1 (x) we obtain that, as x → 0+,
(s + 1)xs n · Q2 (0) z2 F ′ (0)G (0) − F (0)G ′ (0) Q(x) ∼ 2 Q1 (0)
o ′ ′ − Q1 (0) z1 F (0)G (0) − F (0)G (0) (s + 1)xs F ′ (0)G (0)(z2 − z1 ) −sF (0)H (0) √ 2s z1 − z2 s v Γ(−s) x. ∼ Γ(s + 1) 2 4 ∼

Combining the above asymptotics, we get (54). Substituting (52), (53) and (54) to (51), we have (s,v)

(s,v)

(s,v)

(s,v)

AII (ε, z1 )BII (ε, z2 ) − AII (ε, z2 )BII (ε, z1 )  √ 2 Ks (ε v) √  z1 z2 s −1 2(z1 − z2 ) ε , as ε → 0 + . ∼ Γ(−s)Γ(s + 1) v (1)

Finally, by (50), we get the formula for S∞ (0, z1 , z2 ): √  z1 z2 s 1 1 p = N (s,v) (z1 )N Γ(−s)Γ(s + 1) (v + z1 )(v + z2 ) v

(s,v)

(z2 ). 

For α > −1, we denote by Je(α) (x, y) the Bessel kernel, i.e., √ √ ′ √ √ √ ′ √ J ( x) yJ ( y) − J ( y) xJα ( x) α α α Je(α) (x, y) = . 2(x − y)

It is well-known (cf. e.g. [16]) that the Bessel kernel has the following integral representation: Z √ √ 1 1 (α) e Jα ( tx)Jα ( ty)dt. J (x, y) = 4 0 Proposition 4.23. Let m = 1 and −1 < s < 0. Then

e(s+1) (z1 , z2 ). lim Π(s,v) ∞ (z1 , z2 ) = J

v→0+

Moreover, the convergence is uniform as soon as z1 , z2 are in compact subsets of (0, ∞).

38

ALEXANDER I. BUFETOV , YANQI QIU

Proof. Fix −1 < s < 0. It is easy to see that lim N

v→0+

(s,v)

(z) = 0,

and the convergence is uniform for z in compact subset of (0, ∞). By (12) and (18), we have √ √ √ Ks (κ v) − vKs+1 (κ v) (s,v) AII (κ, zi ) = . √ √ √ Js (κ zi ) − zi Js+1(κ zi )

Then apply the asymptotics of Ks , Ks+1 near 0 to get

(s,v) √ √ AII (κ, zi ) √ lim = − zi Js+1 (κ zi ). v→∞ Ks (κ v)

It follows that lim I

v→0+

(s,v)

√ √ Js+1 (κ z1 )Js+1 (κ z2 ) · κ. (κ, z1 , z2 ) = 2

For any 0 < ε < 1, the convergence is uniform as long as κ ∈ [ε, 1] and z1 , z2 in compact subsets of (0, ∞). Hence Z 1 Z 1 √ √ Js+1 (κ z1 )Js+1 (κ z2 ) (s,v) · κdκ. lim I (κ, z1 , z2 )dκ = v→0+ ε 2 ε The above term tends to Z 1 Z √ √ √ √ Js+1 (κ z1 )Js+1(κ z2 ) 1 1 · κdκ = Js+1 ( tz1 )Js+1 ( tz2 )dt 2 4 0 0 uniformly as z1 , z2 in compact subsets of (0, ∞), when ε → 0. It is easy to see that Z ε (s,v) . εs+1. sup I (κ, z , z )dκ 1 2 0 0, the convergences are uniform as long as κ ∈ [ε, 1] and zi ranges compact simple connected subset of C \ {0}. Proof. The proof is similar to that of Proposition 4.4.



Proposition 4.29. In the regime (55), we have o n 3 − s (s,u) (n) (n) (n) (x ) − p (x ) = 23s+ 2 u−1−s zi 2 BIII (κ, zi ) lim n2−3s p(n) κn −1 i κn i n→∞

and

o n (n) (n) (n) (n) (n) (n) (n) (x )p (x ) (x )p (x ) − p lim n3−6s pκ(n) 2 κn −1 1 1 κn −1 2 κn n n→∞ n o s (s,u) (s,u) (s,u) (s,u) =26s+3 u−2−2s (z1 z2 )− 2 BIII (κ, z1 )AIII (κ, z2 ) − BIII (κ, z2 )AIII (κ, z1 ) ,

42

ALEXANDER I. BUFETOV , YANQI QIU (s,u)

where BIII (κ, z) =

(s,u) ∂ A (κ, z), ∂κ III

i.e., √ K (κ u) u 21 K ′ (κ√u) u 23 Ks(3) (κ√u) s s √ √ √ 1 3 (s,u) BIII (κ, z) = Ls (κ u) u 2 L′s (κ u) u 2 L(3) s (κ u) Js (κ√z) z 12 J ′ (κ√z) z 32 Js(3) (κ√z) s

.

Moreover, for any ε > 0, the convergences are uniform as long as κ ∈ [ε, 1] and zi ranges compact simple connected subset of C \ {0}. 

Proof. The proof is similar to that of Proposition 4.7. Now we obtain the following theorem. Theorem 4.30. In the regime (55), we obtain the scaling limit e (n) Φ(s,u) ∞ (z1 , z2 ) := lim Φn (z1 , z2 ) =

n→∞ (s,u) (s,u) AIII (1, z1 )BIII (1, z2 )

(s,u)

(s,u)

− AIII (1, z2 )BIII (1, z1 ) .  (s,u) 2 2(u + z1 )(u + z2 ) · C3 (1) · (z1 − z2 ) (s,u)

For investigating the integral form of the scaling limit Φ∞ , let us put Z 1 1 (n) (n) p0 (t) ≡ 1 and p1 (t) = 1 − t − (n) (1 − t)w bs(n) (t)dt. b h0 −1 (n)

The contribution of p0 to the kernel is q (n) (n) (n) (n) s (n) (n) (n) w bs (x1 )w bs (x2 ) p(n) (z1 z2 ) 2 (2n2 )1−s 0 (x1 )p0 (x2 ) = . · · R 1 (n) (1−t)s b 2n2 (z1 + u)(z2 + u) dt h0 u 2 −t) −1 (1+ 2n2

We note that for −1 < s < 1, we have R1 (1−t)s dt Z 4n2 −1 (1+ u2 −t)2 ts 2n = dt (2n2 )1−s (t + u)2 0 Z ∞ ts n→∞ dt = us−1B(1 + s, 1 − s) = us−1 Γ(1 + s)Γ(1 − s). −−−→ 2 (t + u) 0 (n)

The contribution of p1 to the kernel is q (n) (n) (n) (n) (n) (n) (n) w bs (x1 )w bs (x2 ) p(n) 1 (x1 )p1 (x2 ) · (n) b 2n2 h1 s

(n)

(n)

(n)

(n)

(z1 z2 ) 2 (2n2 )1−s p1 (x1 )p1 (x2 ) = · . (n) b (z1 + u)(z2 + u) h1

THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

For −1 < s < 0, we have 1 (n) b h0

Hence

Z

1

−1

(n)

(1 −

t)w bs(n) (t)dt

(n)

p1 (xi ) = and



Z

4n2 0

It follows that

(n)

(n)

(n)



1

= R∞ 0




ts+1 dt (t+u)2 . R 4n2 ts 2n2 0 (t+u)2 dt 0

R 4n2




1+s u z2 + 1+s u s s .
s 2 y u (y+u) y + 1+s 2 dy s

0

ts+1 dt (t+u)2 ts dt (t+u)2

R 4n2

y − R0 2 4n

(2n2 )1−s p1 (x1 )p1 (x2 ) lim = (n) n→∞ b h z1 +

R 4n2

1  0 zi − R 4n2 2 2n

(n) b h1 = (2n2 )−1−s

(n)

=

43

0

z1 − R∞ 0



,

ts+1 dt (t+u)2 ts dt (t+u)2

R∞

ts+1 0 (t+u)2 dt R ∞ ts dt 0 (t+u)2

 y−

2

 dy.

!

z2 −

R∞

ts+1 0 (t+u)2 dt R ∞ ts 0 (t+u)2 dt

R∞

ts+1 0 (t+u)2 dt R ∞ ts dt 0 (t+u)2

2

ys dy (y+u)2

Definition 4.31. For −1 < s < 1, define a positive function on R∗+ : s

(s,u) M0 (z)

For −1 < s < 0, define (s,u) M1 (z)

:= hR ∞ 0

z2 1 · . := p s−1 u Γ(1 + s)Γ)(1 + s) z + u 1
2 u y + 1+s s (s,u)

we extend the definition of M1

s

ys dy (y+u)2

i1/2

z2 1+s u) · , · (z + s z+u (s,u)

when 0 ≤ s < 1 by setting M1

≡ 0.

The detail proof of the following proposition is long but routine and similar to the proof of Proposition 4.22, so we omit its proof here.

!

44

ALEXANDER I. BUFETOV , YANQI QIU

Proposition 4.32. For −1 < s < 1, we have the following representation (s,u) of Φ∞ (z1 , z2 ): (s,u)

Φ(s,u) ∞ (z1 , z2 ) =M0

(s,u)

(z1 )M0

(s,u)

(s,u)

(z2 ) + M1 (z1 )M1 (z2 ) Z 1 (s,u) (s,u) AIII (κ, z1 )AIII (κ, z2 ) 1 + κdκ.  (s,u) 2 2(z1 + u)(z2 + u) 0 CIII (κ) R EFERENCES

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THE EXPLICIT FORMULAE FOR THE SCALING LIMITS

45

[15] G´abor Szeg˝o. Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. [16] Craig A. Tracy and Harold Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289–309, 1994. [17] A.M. Vershik, A description of invariant measures for actions of certain infinitedimensional groups. (Russian) Dokl. Akad. Nauk SSSR 218 (1974), 749–752. A.B.:I NSTITUT DE M ATH E´ MATIQUES ´ , CNRS, M ARSEILLE SIT E

DE

M ARSEILLE , A IX -M ARSEILLE U NIVER -

S TEKLOV I NSTITUTE OF M ATHEMATICS , M OSCOW I NSTITUTE FOR I NFORMATION T RANSMISSION P ROBLEMS , M OSCOW NATIONAL R ESEARCH U NIVERSITY H IGHER S CHOOL

OF

E CONOMICS , M OSCOW

R ICE U NIVERSITY, H OUSTON TX Y. Q.: I NSTITUT DE M ATH E´ MATIQUES DE M ARSEILLE , A IX -M ARSEILLE U NIVER M ARSEILLE

´, SIT E