Uniform Asymptotics for Orthogonal Polynomials

P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides and X. Zhou

Abstract. We consider asymptotics of orthogonal polynomials with respect to a weight e^{−Q(x)}dx on ℝ, where either Q(x) is a polynomial of even order with positive leading coefficient, or Q(x) = NV (x), where V (x) is real analytic on ℝ and grows sufficiently rapidly as |x| → ∞. We formulate the orthogonal polynomial problem as a Riemann-Hilbert problem following the work of Fokas, Its and Kitaev. We employ the steepest descent-type method for Riemann-Hilbert problems introduced by Deift and Zhou, and further developed by Deift, Venakides and Zhou, in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients and the recurrence coefficients of the orthogonal polynomials. These asymptotics are also used to prove various universality conjectures in the theory of random matrices.

1991 Mathematics Subject Classification: 33D45, 60F99, 15A52, 45E05. Keywords and Phrases: orthogonal polynomials, asymptotics, random matrix theory, universality.

Let w(x)dx = e^{−Q(x)}dx be a measure on the real line. Denote by π_{n}(x, Q) = π_{n}(x) = x^{n} + … the n-th monic orthogonal polynomial with respect to the measure, and by p_{n}(x, Q) = p_{n}(x) = γ_{n}π_{n}(x), γ_{n} > 0, the normalized n-th orthogonal polynomial, or simply the n-th orthogonal polynomial, i.e.

∫_{ℝ} p_{n}(x)p_{m}(x)e^{−Q(x)}dx = δ_{n,m}  , n, m ∈ ℕ.  (1)

Furthermore, denote by (a_{n})_{n∈ℕ}, (b_{n})_{n∈ℕ} the coefficients of the associated three term recurrence relation, namely, xp_{n}(x) = b_{n}p_{n+1}(x) + a_{n}p_{n}(x) + b_{n−1}p_{n−1}(x), n ∈ ℕ, and denote by x_{1,n} > x_{2,n} > … > x_{n,n} the roots of p_{n}.

In [8], the authors considered the case where

Q(x) = ∑_{k=0}^{2m} q_{k}x^{k},  q_{2m} > 0, m > 0,  (2)

is a polynomial of even degree with a positive leading coefficient, and in [7] the case where

Q(x) = NV (x), V (x) is real analytic on ℝ,
and  \frac{V(x)}{log(x^{2} + 1)} → ∞  as x → ∞.

(3)

In [8], the authors are concerned with the asymptotics as n → ∞ of the leading coefficient γ_{n}, the recurrence coefficient a_{n}, b_{n} and the zeros x_{jn}, as well as Plancherel-Rotach-type asymptotics for the orthogonal polynomials p_{n}, i.e asymptotics for p_{n}(zc_{n}+d_{n}) uniformly for all z ∈ ℂ, where c_{n}, d_{n} are certain quantities related to the so-called Mhaskar-Rahmanov-Saff numbers (see (5) below). The name “Plancherel-Rotach” refers to [17] in which the authors prove asymptotics of this type for the classical case of Hermite polynomials. In [7], the authors are concerned with the asymptotics of γ_{n}, a_{n}, b_{n} and p_{n}(z; NV) in the case c^{−1}N ≤ n ≤ cN for some c > 1, as N → ∞. These asymptotics are crucial ingredients in proving a variety of universality conjectures in random matrix theory (see [7]).

Due to the page restrictions in these Proceedings, we limit our considerations to a description of the results in [8]. Plancherel-Rotach-type asymptotics for polynomial orthogonal with respect to exponential weights of the above type, play a central role in various questions of weighted approximation on the line (see e.g. [15]). In order to prove our results we use a reformulation of the orthogonal polynomial problem as a Riemann-Hilbert problem, due to Fokas, Its and Kitaev [13, 14] (see below). This Riemann-Hilbert problem is then analyzed in turn asymptotically using the non-commutative steepest-descent method introduced by Deift and Zhou in [11], and further developed in [12] and [9], and placed eventually in a general form by Deift, Venakides and Zhou in [10]. In [8], and particularly in [7], a basic role is played by the results on the equilibrium measure (see below) obtained by Deift, Kriecherbauer and Ken McLaughlin in [5]. In this paper we will only have the opportunity to give a very rough sketch of the steepest descent method: full details can be found in [8]. For the case of varying weights e^{−NV (x)}dx, we must, alas, refer the reader to [7], for both a detailed description of the results as well as their proofs, and the connection to random matrix theory. The methods in [7] are similar to those in [8], but require additional technical considerations. In the special case where V is an even quartic polynomial, the results in [7] should be compared with the results of Bleher and Its [2], who were the first to use the steepest-descent method in [11] to study the asymptotics of orthogonal polynomials via a Riemann-Hilbert problem. Some of the results in [7] and in [8] were announced in [6].

There is a vast literature on asymptotic questions for orthogonal polynomials. The list of researchers who have made important contributions close to the results of [7] and [8], includes, in addition to Plancherel and Rotach, and Bleher and Its, Bauldry, Chen, Criscuolo, Della Vechia, Geronimo, Ismail, Lubinsky, Magnus, Maskar, Mastroiani, Mate, Nevai, Rahmanov, Saff, Sheen, Totik and Van Assche, but there are many others. Again, we do not have the opportunity to describe their work in any detail. Fortunately there is an excellent review [15]: also, a detailed description of the work of the above authors related to the present paper is given in [8].

Henceforth we will assume that the potential Q(x) is of the form (2). The statement of our results involves the n-th Mhaskar-Rahmanov-Saff numbers (in short: MRS-numbers [16], [18]) α_{n}, β_{n} which can be determined from the equations

\frac{1}{2π} ∫_{α_{n}}^{β_{n}} \frac{Q′(t)(t − α_{n})}{\sqrt{(β_{n} − t)(t − α_{n})}}dt = n,  \frac{1}{2π} ∫_{α_{n}}^{β_{n}} \frac{Q′(t)(β_{n} − t)}{\sqrt{(β_{n} − t)(t − α_{n})}}dt = −n, (4)

and in particular the interval [α_{n}, β_{n}] whose width and midpoint are given by

c_{n} := \frac{β_{n} − α_{n}}{2} ,  d_{n} := \frac{β_{n} + α_{n}}{2} .  (5)

For the weights under consideration it is straightforward to prove the existence of the MRS-numbers for sufficiently large n. Moreover, they can be expressed in a power series in n^{− \frac{1}{2m}}. We obtain

c_{n} = n^{\frac{1}{2m}} ∑_{l=0}^{∞} c^{(l)}n^{− \frac{l}{2m}} ,  d_{n} = ∑_{l=0}^{∞} d^{(l)}n^{− \frac{l}{2m}} ,  (6)

where the coefficients c^{(l)}, d^{(l)} can be computed explicitly. From now on we will assume that n is sufficiently large for (6) to hold.

Statement of Results

To simplify the analysis, we normalize the interval [α_{n}, β_{n}] to be [−1, 1] by making the linear change of variable

λ_{n} : ℂ → ℂ : z ↦ c_{n}z + d_{n},  (7)

which takes the interval [−1, 1] onto [α_{n}, β_{n}], and we work with the function

V_{n}(z) := \frac{1}{n}Q(λ_{n}(z)).  (8)

The function V_{n} is again a polynomial of degree 2m with leading coefficient (mA_{m})^{−1} > 0, whereas all other coefficients tend to zero as n tends to ∞.

We present our results in terms of the well-known equilibrium measure μ_{n} (see e.g. [19]) with respect to V_{n} which is defined as the unique minimizer in M_{1}(ℝ) = {probability measures on ℝ} of the functional

I^{V_{n}} : M_{1}(ℝ) → (−∞, ∞] : μ ↦ ∫_{ℝ^{2}} log |x − y|^{−1}dμ(x)dμ(y) + ∫_{ℝ} V_{n}(x)dμ(x). (9)

The equilibrium measure and the corresponding variational problem emerge naturally in our asymptotic analysis of the Riemann-Hilbert problem. The minimizing measure is given by

dμ_{n}(x) = \frac{1}{2π} \sqrt{1 − x^{2}}h_{n}(x)1_{[−1,1]}(x)dx,  (10)

where 1_{[−1,1]} denotes the indicator function of the set [−1, 1] and h_{n} is a polynomial of degree 2m − 2,

h_{n}(x) = ∑_{k=0}^{2m−2} h_{n,k}x^{k},  (11)

and the coefficients h_{n,k} can be expanded in an explicitly computable power series in n^{− \frac{1}{2m}}.

Finally, to state our first theorem, we define

l_{n} := \frac{1}{π} ∫_{−1}^{1} \sqrt{1 − t^{2}}h_{n}(t) log |t|dt − V_{n}(0),  (12)

which also has an explicitly computable power series in n^{− \frac{1}{2m}}.

Asymptotics of the leading and recurrence coefficients of the orthogonal polynomials p_{n}

Theorem 13. In the above notation we have

\begin{matrix}γ_{n}\sqrt{πc_{n}^{2n+1}e^{nl_{n}}} = 1 − \frac{1}{n} (\frac{4h_{n}(1) − 3h′_{n}(1)}{48h_{n}(1)^{2}} + \frac{4h_{n}(−1) + 3h′_{n}(−1)}{48h_{n}(−1)^{2}}) + O (\frac{1}{n^{2}} ) , \\ \frac{b_{n−1}}{c_{n}} = \frac{1}{2} + O (\frac{1}{n^{2}} ) , a_{n} = d_{n} + \frac{c_{n}}{2n} (\frac{1}{h_{n}(1)} − \frac{1}{h_{n}(−1)} + O (\frac{1}{n})) . \end{matrix} (14)

In all three cases there are explicit integral formulae for the error terms which all have an asymptotic expansion in n^{− \frac{1}{2m}}, e.g. O (\frac{1}{n}) = \frac{1}{n} (κ_{0} + κ_{1}n^{− \frac{1}{2m}} + …). The coefficients of these expansions can be computed via the calculus of residues by purely algebraic means.

Next we will state the Plancherel-Rotach type asymptotics of the orthogonal polynomials p_{n}, i.e. the limiting behavior of the rescaled n-th orthogonal polynomial p_{n}(λ_{n}(z)), as n tends to infinity and z ∈ ℂ remains fixed. We will give the leading order behavior and produce error bounds which are uniform in the entire complex plane ℂ. Notation: In the following, (·)^{α}, α ∈ ℝ, denotes the principal branch of the α^{th} root. On the other hand, we will reserve the notation \sqrt{a} for nonnegative numbers a, and we always take \sqrt{a} nonnegative: thus \sqrt{1 − x^{2}}, −1 ≤ x ≤ 1 in (10) is positive.

Plancherel-Rotach Asymptotics

We state our second theorem in terms of the function

ψ_{n} : ℂ \ ((−∞, −1] ∪ [1, ∞)) → ℂ : z ↦ \frac{1}{2π} (1 − z)^{1/2}(1 + z)^{1/2}h_{n}(z). (15)

The function ψ_{n} is an analytic extension of the density of μ_{n} on (−1, 1) to ℂ \ ((−∞, −1] ∪ [1, ∞)) and is thus closely linked to the equilibrium measure (cf. (10)). We show that there exist analytic functions f_{n}, f̃_{n} in a neighborhood of 1, respectively −1, satisfying

(−f_{n}(z))^{3/2} = −n\frac{3π}{2} ∫_{1}^{z} ψ_{n}(y)dy, for |z − 1| small, z ∉ [1, ∞).

(f̃_{n}(z))^{3/2} = n\frac{3π}{2} ∫_{−1}^{z} ψ_{n}(y)dy, for |z + 1| small, z ∉ (−∞, −1].

(16)

As p_{n}(z) = \overline{p_{n}(z̄)}, it is sufficient to describe the asymptotics of p_{n}(c_{n}z + d_{n}) in the closed upper half plane \overline{ℂ}_{+}. Depending on a small parameter δ, we divide \overline{ℂ}_{+} into six closed regions, as shown in Figure 17 below. We only describe the asymptotics in A_{δ}, C_{1,δ}, C_{2,δ} and B_{δ}. The asymptotics in D_{j,δ}, j = 1, 2, is of a similar form to that in C_{j,δ}, j = 1, 2 respectively, with f̃_{n} replacing f_{n}. Let Ai(z) denote the Airy function [1, 10.4].

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Figure 17. Different asymptotic regions for p_{n}(c_{n}z + d_{n}) in \overline{ℂ_{±}}.

Theorem 18. There exists a δ_{0} such that for all 0 < δ ≤ δ_{0} the following holds (see Figure 17):

(i) For z ∈ A_{δ}:

p_{n}(c_{n}z + d_{n})e^{− \frac{1}{2} Q(c_{n}z+d_{n})} = \sqrt{\frac{1}{4πc_{n}}} exp (−nπi ∫_{1}^{z} ψ_{n}(y)dy)  (19)

× (\frac{(z − 1)^{1/4}}{(z + 1)^{1/4}} + \frac{(z + 1)^{1/4}}{(z − 1)^{1/4}} ) (1 + O (\frac{1}{n})) .

(ii) For z ∈ B_{δ}:

p_{n}(c_{n}z + d_{n})e^{− \frac{1}{2} Q(c_{n}z+d_{n})} = \sqrt{\frac{2}{πc_{n}}} (1 − z)^{−1/4}(1 + z)^{−1/4}  (20)

× \{cos (nπ ∫_{1}^{z} ψ_{n}(y)dy + \frac{1}{2} arcsin z) (1 + O (\frac{1}{n}))

+ sin (nπ ∫_{1}^{z} ψ_{n}(y)dy + \frac{1}{2} arcsin z) O (\frac{1}{n})\} .

(iii) For z ∈ C_{1,δ}:
p_{n}(c_{n}z + d_{n})e^{− \frac{1}{2} Q(c_{n}z+d_{n})}  (21)

=\sqrt{\frac{1}{c_{n}}} \{(\frac{(z + 1)^{1/4}}{(z − 1)^{1/4}} (f_{n}(z))^{1/4} Ai (f_{n}(z))) (1 + O (\frac{1}{n}))

− (\frac{(z − 1)^{1/4}}{(z + 1)^{1/4}} (f_{n}(z))^{−1/4} Ai′ (f_{n}(z))) (1 + O (\frac{1}{n}))\} . (22)
(iv) For z ∈ C_{2,δ}:

p_{n}(c_{n}z + d_{n})e^{− \frac{1}{2} Q(c_{n}z+d_{n})} = \sqrt{\frac{1}{c_{n}}} \{\frac{(z + 1)^{1/4}}{(z − 1)^{1/4}} (f_{n}(z))^{1/4} Ai(f_{n}(z))

− \frac{(z − 1)^{1/4}}{(z + 1)^{1/4}} (f_{n}(z))^{−1/4} Ai′(f_{n}(z))\} (1 + O (\frac{1}{n})) .  (23)

All the error terms are uniform for δ ∈ compact subsets of (0, δ_{0}] and for z ∈ X_{δ}, where X ∈ {A, B, C_{1}, C_{2}}. There are integral formulae for the error terms from which one can extract an explicit asymptotic expansion in n^{− \frac{1}{2m}}.

Remarks:

(a) Some of the expressions in Theorem 18 are not well defined for all z ∈ ℝ (see e.g. (z − 1)^{1/4}, ∫_{1}^{z} ψ_{n}(y)dy). In these cases we always take the limiting expressions as z is approached from the upper half-plane.

(b) The function arcsin is defined as the inverse function of

sin : \{z ∈ ℂ : |Re(z)| < \frac{π}{2} \} → ℂ \ ((−∞, −1] ∪ [1, ∞)) .

Asymptotic Location of the Zeros

In order to state our result on the location of the zeros, we denote the zeros of the Airy function Ai by 0 > −ι_{1} > −ι_{2} > …. Recall that the all the zeros of Ai lie in (−∞, 0), so that there exists a largest zero −ι_{1} < 0. Furthermore, note that [−1, 1] ∋ x ↦ ∫_{x}^{1} ψ_{n}(t)dt ∈ [0, 1] is bijective and we define its inverse function to be ζ_{n} : [0, 1] ↦ [−1, 1].

Theorem 24. The zeros x_{1,n} > x_{2,n} > … > x_{n,n} of the n-th orthogonal polynomials p_{n} satisfy the following asymptotic formulae:

(i) Fix k ∈ ℕ. Then

\frac{x_{k,n} − d_{n}}{c_{n}} = 1 − (\frac{2}{h_{n}(1)^{2}} )^{1/3} \frac{ι_{k}}{n^{2/3}} + O (\frac{1}{n}) , as n → ∞,  (25)

and

\frac{x_{n−k,n} − d_{n}}{c_{n}} = −1 + (\frac{2}{h_{n}(−1)^{2}} )^{1/3} \frac{ι_{k}}{n^{2/3}} + O (\frac{1}{n}) , as n → ∞.  (26)

(ii) There exist constants k_{0}, C > 0, such that for all k_{0} ≤ k ≤ n − k_{0} the following holds:

\frac{x_{k,n} − d_{n}}{c_{n}} ∈ (ζ_{n} (\frac{6k − 1}{6n}) , ζ_{n} (\frac{6k − 5}{6n})) .  (27)

|\frac{x_{k,n} − d_{n}}{c_{n}} − ζ_{n} (\frac{6k − 3}{6n} + \frac{1}{2πn} arcsin (ζ_{n} (k/n)))| ≤ \frac{C}{n^{2}[α(1 − α)]^{4/3}} , (28)

where α := k/n.

(iii) There exists a constant C_{1} > 0 such that

\frac{1}{C_{1}} < \frac{x_{k,n} − x_{k+1,n}}{c_{n}[nk(n − k)]^{1/3}} < C_{1} for all 1 ≤ k ≤ n − 1.  (29)

Remarks:

(a) Using the asymptotic expansion for the error terms in Theorem 18 one can of course approximate the k-th zero x_{k,n} of the orthogonal polynomial p_{n} to arbitrary accuracy.

(b) Note that the error term in (28) is at most of order O(n^{−2/3}). Furthermore it is obvious that for any compact subset K of (0, 1), there exists a constant C_{K}, such that the error term in (28) is bounded by C_{K}/n^{2}, as long as α = k/n ∈ K.

As noted earlier, our approach to the asymptotic problem for orthogonal polynomials, is based on the reformulation of the orthogonal polynomial problem as a Riemann-Hilbert problem due to Fokas, Its and Kitaev (see [13], [14]: a specialized version appeared also in [4]).

A general reference for Riemann-Hilbert problems is, for example, [3]. Let Σ be an oriented contour in ℂ.

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+ −  + − + + − + +−  +−  −

Figure 30.

As indicated in the Figure, the (+)-side (resp, (−)-side) of the contour lies to left (resp, right) as one moves along the contour in the direction of the orientation. Let v be a given map from Σ to Gl(k, ℂ). We say that m = m(z) is a solution of the Riemann-Hilbert problem (Σ, v) if

• m(z) is analytic in ℂ − Σ,

• m_{+}(z) = m_{−}(z)v(z) , z ∈ Σ,

where m_{±}(z) = lim_{z′→zz′∈(±)−side} m(z′). The matrix v is called the jump matrix for the Riemann-Hilbert problem. If in addition

• m(z) → I as z → ∞,

we say that the the Riemann-Hilbert problem is normalized at infinity.

Theorem 31. ([13, 14]) Let w : ℝ → ℝ_{+} denote a function with the property that w(s)s^{k} belongs to the Sobolev space H^{1}(ℝ) for all k ∈ ℕ. Suppose furthermore that n is a positive integer. Then the Riemann-Hilbert problem on Σ = ℝ, oriented from −∞ to +∞,

Y : ℂ \ ℝ → ℂ^{2×2} is analytic , Y_{+}(s) = Y_{−}(s) ( \begin{matrix}1 & w(s) \\ 0 & 1 \end{matrix} ) for s ∈ ℝ,

Y (z) ( \begin{matrix}z^{−n} & 0 \\ 0 & z^{n} \end{matrix} ) = I + O (\frac{1}{|z|}) , as |z| → ∞,

(32)

has a unique solution, given by

Y (z) = ( \begin{matrix}π_{n}(z) & ∫ℝ \frac{π_{n}(s)w(s)}{s−z} \frac{ds}{2πi} \\ −2πiγ_{n−1}^{2}π_{n−1}(z) & ∫_{ℝ} \frac{−γ_{n−1}^{2}π_{n−1}(s)w(s)}{s−z}ds \end{matrix} ) ,  (33)

where π_{n} denotes the n-th monic orthogonal polynomial with respect to the measure w(x)dx on ℝ and γ_{n} > 0 denotes the leading coefficient of the n-th orthogonal polynomial p_{n} = γ_{n}π_{n}. Furthermore, there exist Y_{1}, Y_{2} ∈ ℂ^{2×2} such that

Y (z) ( \begin{matrix}z^{−n} & 0 \\ 0 & z^{n} \end{matrix} ) = I + \frac{Y_{1}}{z} + \frac{Y_{2}}{z^{2}} + O (\frac{1}{|z|^{3}} ) , as |z| → ∞,

and γ_{n−1} = \sqrt{(Y_{1})_{21}/ − 2πi}, γ_{n} = 1/\sqrt{−2πi(Y_{1})_{12}},

a_{n} = (Y_{1})_{11} + (Y_{2})_{12}/(Y_{1})_{12}, b_{n−1} = \sqrt{(Y_{1})_{21}(Y_{1})_{12}},

(34)

where a_{n}, b_{n} are the recurrence coefficients associated to the orthogonal polynomials p_{n}.

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Figure 35. The contour Σ_{S}.

Documenta Mathematica · Extra Volume ICM 1998 · III · 491–501

We are interested in the case where w(x) = e^{−Q(x)}, and Q(x) satisfies (2). Theorem 31 converts the problem of computing the asymptotics of γ_{n}, a_{n}, b_{n}, … into a problem of computing the asymptotics of the Riemann-Hilbert problem (32) as n → ∞. As indicated, this is achieved by using the steepest descent method for Riemann-Hilbert problems introduced in [11], and further developed in [12]. We conclude with a brief sketch of the method, which involves a sequence of transformations of the Riemann-Hilbert problem:

(i) rescaling: Y → U_{n}(z) ≡ ( \begin{matrix}c_{n}^{−n} & 0 \\ 0 & c_{n}^{n} \end{matrix} ) Y (c_{n}z + d_{n}), where c_{n}, d_{n} are

related to the MRS-numbers as in (5).

(ii) introduction of the “g-function” which is the analog for the Riemann-Hilbert problem of the phase function of linear WKB theory: U → T (z) ≡

e^{−nlσ_{3}/2}U(z)e^{−n(g(z)−l/2)σ_{3}} where σ_{3} is the Pauli matrix ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and l = l_{n}

is given in (12). The function g(z) is analytic in ℂ \ ℝ, has asymptotics g(z) ∼ log z as z → ∞ and is uniquely determined as in [10] by requiring that g_{±}(z) ≡ lim_{ϵ→0^{+}} g(z ± iϵ) satisfy certain equalities and inequalities (“Phase Conditions”) on ℝ. A simple computation shows that T (z) is the solution of the following Riemann-Hilbert problem, normalized at infinity:

• T (z) is analytic in ℂ \ ℝ,

• T_{+}(z) = T_{−}(z) ( \begin{matrix}e^{−n(g_{+}(z)−g_{−}(z))} & e^{n(g_{+}(z)+g_{−}(z)−V_{n}(z)−l)} \\ 0 & e^{n(g_{+}(z)−g_{−}(z))} \end{matrix} ) for z ∈ ℝ,

• T (z) = I + O(\frac{1}{|z|}) as z → ∞.

(iii) involves a factorization of the jump matrix and a deformation of the contour: T → S. The 2×2 matrix function S = S(z) solves a Riemann-Hilbert problem on a contour of type Σ_{S} as in Figure 35. Now the Phase Conditions in (ii) are chosen precisely to ensure that the jump matrix v_{S} for S on Σ_{1}, Σ_{3}, Σ_{4} and Σ_{5}, converges exponentially to the identity matrix as n → ∞, whereas v_{S} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) on Σ_{2} = [−1, 1]. Thus as n → ∞, we expect that S converges to the solution of the simple Riemann-Hilbert problem (Σ_{2} = [−1, 1], v = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )), which may be solved in turn in terms of elementary radicals.

The final step (iv) involves the construction, following [12], of a parametrix for S at the points of self-intersection {−1, 1} of Σ_{S}: S → R. Although v_{S} → I on Σ_{1}∪Σ_{3}∪Σ_{4}∪Σ_{5}, the convergence is not uniform and is slower and slower near 1 and −1. This is the central analytical difficulty in the method, and requires delicate consideration. The parametrix for S is chosen so that R solves a Riemann-Hilbert problem on an extended contour Σ_{R} ⊃ Σ_{S} with a jump matrix v_{R} satisfying ∥v_{R} − I∥_{L^{∞}(Σ_{R})} → 0 as n → ∞. By standard Riemann-Hilbert methods, R can then be solved in terms of a Neumann series, and retracing the steps R → S → T → U → Y, we obtain the asymptotics for γ_{n}, a_{n}, b_{n}, x_{kn} and p_{n}(c_{n}z + d_{n}) as advertised in Theorem 13, 18 and 24.

Finally we note that it is a remarkable piece of luck that the phase condition in (ii) above can be expressed simply in terms of the equilibrium measure dμ_{n} corresponding to V_{n}(z) as in (9), (10) above. Indeed, if we set g(z) = ∫ log(z − x)dμ_{n}, then it turns out that the Euler-Lagrange variational equations for μ_{n}, the minimizing measure in (9), are equivalent to the desired phase condition on g. In this way we construct the g-function in terms of the equilibrium measure.

Acknowledgements. The authors would like to thank Alexander Its and Peter Sarnak for many useful conversations on orthogonal polynomials. The authors would also like to thank Alexander Its and Pavel Bleher for making available to them their preprint [2] at an early stage. Percy Deift was supported in part by NSF grant # DMS-9500867. Thomas Kriecherbauer was supported in part by DFG grant # Kr 1673/2-1. Kenneth T-R McLaughlin was supported in part by NSF postdoctoral fellowship grant # DMS-9508946. Stephanos Venakides was supported in part by NSF grant # DMS-9500623 and ARO grant # DAAH 04-96-1-0157. Xin Zhou was supported in part by NSF grant # DMS-9706644.

References

[1] M. Abramowitz, and I.A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1968.

[2] P. Bleher and A. Its. Asymptotics of Orthogonal Polynomials and Universality in Matrix Models. Preprint, 1996.

[3] K. Clancey and I. Gohberg. Factorization of matrix functions and singular integral operators. Operator Theory, 3, Birkhäuser Verlag, Basel, (1981).

[4] P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou. The Toda Rarefaction Problem. Comm. Pure Appl. Math., 49, 35-83, (1996).

[5] P. Deift, T. Kriecherbauer, and K. T-R McLaughlin. New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field. To appear, J. Approx. Theory.

[6] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides and X. Zhou. Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights. International Mathematics Research Notices 16, 759-782, (1997).

[7] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides and X. Zhou. Uniform Asymptotics for Polynomials Orthogonal with respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory. Preprint (1998).

[8] P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides and X. Zhou. Strong Asymptotics for Orthogonal Polynomials with respect to Varying Exponential Weights. Preprint (1998).

[9] P. Deift, S. Venakides, and X. Zhou. The collisionless Shock Region for the Long-time Behavior of Solutions of the KdV Equation. Comm. Pure and Appl. Math., 47, 199-206 (1994).

[10] P. Deift, S. Venakides, and X. Zhou. New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann - Hilbert problems. Intl. Math. Res. Notes, No.6, 285-299 (1996).

[11] P. Deift and X. Zhou. A steepest descent method for oscillatory Riemann - Hilbert problems. Asymptotics for the mKdV equation, Ann. of Math. 137, 295-370, (1993).

[12] P. Deift and X. Zhou. Asymptotics for the Painlevé II equation. Comm. Pure and Appl. Math. 48, 277-337, (1995).

[13] A.S. Fokas, A.R. Its and A.V.Kitaev. Isomonodromic approach in the theory of two-dimensional quantum gravity. Usp. Matem. Nauk, 45,135-136, (1990) (In Russian).

[14] A.S.Fokas, A.R. Its and A.V. Kitaev. Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys., 142, 313-344, (1991).

[15] D. S. Lubinsky. An Update on Orthogonal Polynomials and Weighted Approximation on the Real Line. Acta Applicandae Mathematicae, 33, 121-164, (1993).

[16] H.N. Mhaskar and E.B. Saff. Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc., 285, 203-234, (1984).

[17] M. Plancherel, and W. Rotach. Sur les valeurs asymptotiques des polynomes d’Hermite H_{n}(x) = (−1)^{n}e^{x^{2}/2}d^{n}(e^{−x^{2}/2})/dx^{n}. Commentarii Mathematici Helvetici, 1, 227-254, (1929).

[18] E.A. Rachmanov. On asymptotic properties of polynomials orthogonal on the real axis. Mat. Sb., 119, 163-203 (1982). English Transl.: Math. USSR - Sb., 47, (1984).

[19] E.B. Saff and V. Totik. Logarithmic Potentials with External Fields. Springer-Verlag, New York, 1997.

P. Deift

Courant Institute of Mathematical Sciences, New York, New York, USA deift@math1.cims.nyu.edu T. Kriecherbauer University of Munich, Munich, Germany, and Courant Institute of Mathematical Sciences, New York, New York, USA tkriech@rz.mathematik. uni-muenchen.de

K. T-R McLaughlin

Princeton University, Princeton, New Jersey, USA, and University of Arizona, Tucson, Arizona, USA mcl@math.arizona.edu S. Venakides and X. Zhou Duke University, Durham, North Carolina, USA ven@math.duke.edu zhou@math.duke.edu 502

Documenta Mathematica · Extra Volume ICM 1998 · III ·