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(n) pl,β (λ1, ..., λl)= pn,β(λ1, ...λl, λl+1, ..., λn)dλl+1...dλn (1.4) Rn−l Z κ in the scaling limit, when λi = λ0 +si/n (i = 1,...,l), and κ is a constant, depending on the behavior of the limiting density ρ(λ) in a small neighborhood of λ0 of the limiting spectrum σ. If ρ(λ ) = 0, then κ = 1, if ρ(λ ) = 0 and ρ(λ) λ λ α, then κ = 1/(1 + α). The 0 6 0 ∼ | − | universality conjecture states that the scaling limits of all marginal densities are universal, i.e. do not depend on V . In the case of unitary ensembles all marginal densities can be expressed (see [12]) in terms of the unique function Kn,2(λ, µ). (n l)! p(n)(λ , ..., λ )= − det K (λ , λ ) l . (1.5) l,β 1 l n! { n,2 j k }j,k=1 This function has the form n−1 (n) (n) Kn,2(λ, µ)= ψl (λ)ψl (µ) (1.6) Xl=0 and is known as a reproducing kernel of the orthonormalized system

ψ(n)(λ) = exp nV (λ)/2 p(n)(λ), l = 0, ..., (1.7) l {− } l in which p(n) n are orthogonal polynomials on R associated with the weight w (λ) = { l }l=0 n e−nV (λ) i.e., (n) (n) pl (λ)pm (λ)wn(λ)dλ = δl,m. (1.8) Z Hence, the problem to study marginal distributions is replaced by the problem to study the behavior of the reproducing kernel Kn(λ, µ) in the scaling limit. This problem was solved in many cases. For example, in the bulk case (ρ(λ ) = 0) it 0 6 was shown in [13], that for a general class of V (the third derivative is bounded in the some neighborhood of λ0)

1 (0) lim Kn,2(λ0 + s1/nρ(λ0), λ0 + s2/nρ(λ0)) = ∞,2(s1,s2), n→∞ nρ(λ0) K where (s ,s ) is a universal sin-kernel K0 1 2 sin π(s s ) (0) (s s )= 1 − 2 . (1.9) K∞,2 1 − 2 π(s s ) 1 − 2 This result for the case of real analytic V was obtained also in [6]. For unitary ensembles it is also possible to study (see [6]) the edge universality, i.e. the case when λ is the edge point of the spectrum and ρ(λ) λ λ 1/2, as λ λ . There are 0 ∼ | − 0| ∼ 0 also results on the extreme point universality (double scaling limit). This means universality of the limiting kernel in the case when ρ(λ) (λ λ )2, as λ λ . See [4] for the result for ∼ − 0 ∼ 0 real analytic potentials and [18] for a general case. For orthogonal ensembles (β = 1 ) the situation is more complicated. Instead of (1.6) we have to use 2 2 matrix kernel × S (λ, µ) S d(λ, µ) K (λ, µ)= n n . (1.10) n,1 IS (λ, µ) ǫ(λ µ) S (µ, λ)  n − − n  2 Here Sn(λ, µ) is some scalar kernel (see (1.24)) below), d denotes the differentiating, ISn(λ, µ) can be obtained from Sn by some integration procedure and ǫ(λ) is defined in (1.22). Similarly to the unitary case all marginal densities can be expressed in terms of the kernel Kn1 (see [21]), e.g. 1 ρ (λ)= p(n)(λ)= TrK (λ, λ), n 1,1 2n n,1 and 1 p(n)(λ, µ)= [TrK (λ, λ)TrK (µ,µ) 2Tr (K (λ, µ)K (µ, λ))] . 2,1 4n(n 1) n,1 n,1 − n,1 n,1 − The matrix kernel (1.10) was introduced first in [10] for circular ensemble and then in [12] for orthogonal ensembles. The scalar kernels of (1.10) could be defined in principle in terms of any family of polynomials complete in L2(R, wn) (see [21]), but usually the families of skew orthogonal polynomials were used (see [12] and references therein). Unfortunately, for general weights the properties of skew orthogonal polynomials are not studied enough. Hence, using of skew orthogonal polynomials for V of a general type rises serious technical difficulties. In the paper [8] a new approach to this problem was proposed. It is based on the result of [21], which allows to express the kernel Sn(λ, µ) in terms of the family of orthogonal (0) polynomials (1.8). Using the representation of [21], it was shown that Sn(λ, µ) K∞,2(λ, µ), (0) → where K∞,2(λ, µ) is defined by (1.9). The same approach was used in [9] to prove the edge universality. Unfortunately, the papers [8] and [9] deal only with the case, when (in our 2m −1/2m 2m−2 notations) V (λ)= λ + n a2m−2λ + . . . . But since, like usually (see [13], [15]), the −1/2m 2m−2 small terms n a2m−2λ + . . . have no influence on the limiting behavior of Kn(λ, µ), this result in fact proves universality for the case of monomial V (λ) = λ2m. In the papers [19, 20] the bulk and the edges universality were studied for the case of V being an even quatric polynomial. In the present paper we prove universality in the bulk of the spectrum for any real analytic V with one interval support. Let us state our main conditions.

C1. V (λ) satisfies (1.2) and is an even analytic function in

Ω[d , d ]= z : 2 d z 2+ d , z d , d , d > 0. (1.11) 1 2 { − − 1 ≤ℜ ≤ 1 |ℑ |≤ 2} 1 2 C2. The support σ of IDS of the ensemble consists of a single interval: