Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and $\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and $x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$ the $\beta$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.

with respect to the classical weight multiplied by (1 − χ (−a,a) (x)) instead of writing such quantities as a Fredholm determinant det(I − K (−a,a) ). Here χ (−a,a) (x) is characteristic function of the interval (a, a).
In our approach, the Hankel determinants is expressed as the product of the square of the L 2 norms, h k (a), of the non − standard orthogonal polynomials namely, n−1 k=0 h k (a). Based on the ladder operators adapted to the non − standard orthogonal polynomials, and from the associated supplementary conditions and a sum-rule, satisfied by certain rational functions (depending on the degree), a series of difference and differential equations can be derived to give a description of P(a, n). For detailed descriptions and applications of such formalism, see for example, [3], [4], [6], [7], [8], [11], [14], and [15].
In [5], such an approach was taken to study the gap probability problem for the Gaussian unitary ensembles (the symmetric situation), namely, the probability that the interval J := (−a, a) is free of eigenvalues. Unfortunately the authors have made a mistake: One term was missed in an equation obtained from the sum-rule. We present the correct version here, but not the derivation and refer the readers to algebraically more complicated case where the back-ground weight is the symmetric Beta density (1 − x 2 ) β , x ∈ [−1, 1] [34]. That is, we study polynomials orthogonal with respect to the deformed weight (1 − x 2 ) β (1 − χ (−a,a) (x)), a < 1, where for convenience we take β > 0. Here and what follows, χ (−a,a) (x) is the characteristic (or the indicator)function of the interval (−a, a), namely χ (−a,a) (x) = 1 if x ∈ (−a, a) and χ (−a,a) (x) = 0, if x / ∈ (−a, a). We note that our approach is entirely elementary, up to some distributional objects. If θ(x) is the step function, which takes value 1 if x > 0 and 0 if x ≤ 0, then d dx θ(x) = δ(x), the Dirac Delta. In the case of the symmetric Jacobi unitary ensembles generated by the weight (1−x 2 ) β , |x| < 1.
The probability that a gap (−a, a), |a| < 1, is formed, can also be found via an "interval doubling", exploiting the parity of the orthogonal polynomials, to the arrive at the Hankel determinants generated by the shifted Jacobi weight x α (1 − x) β , over the interval [t, 1]. Although the asymptotic expansion of the large gap probability can be relatively easily obtained, the determination of the constant of integration is not that straightforward.
In our derivation, we will make use of Dyson's Coulomb fluid approximation [19]. We give a Here v(M) is a matrix function [24] defined via Jordan canonical form and vol(dM) is called the volume element [25]. Under an eigenvalue-eigenvector decomposition, the joint probability density function of the eigenvalues {x k } n k=1 of this ensemble is given by [30] 1 x k w(x)dx, k = 0, 1, 2, · · · .
The normalization constant D n can be evaluated as the determinant of the Hankel (or moment) matrix (see [35]), i.e.
If we interpret {x k } n k=1 as the positions of n identically charged particles, then, for sufficiently large n, the particles can be approximated as a continuous fluid with a density ρ(x). We assume v(x) := log w(x) is convex, so that ρ(x) is supported on a single interval (a, b). Note that (a, b) has nothing to do with (A, B). See [10] for a detailed analysis. Such ρ(x) is determined by minimizing the functional See Dyson's works [19]. According to Frostman's lemma [37], the equilibrium density ρ(x) satisfies where A is the Lagrange multiplier that fixes the condition b a ρ(x)dx = n. The derivative of this equation with respect to x gives rise to the singular integral equation where P denotes the Cauchy principal value. According to the theory of singular integral equations [33], we find Based on the above Coulomb fluid interpretation and the notion of linear statistics, it is proved in [12], for sufficiently large n, that the monic polynomials orthogonal with respect to w( can be approximated as follows: This is a simpler representation for e −S 1 : This paper is organized as follows. In section 2, we give a quick summery of what was know regarding the gap probability of GUE, and finish with an elementary identity expressing the desired Hankel determinant as the products of Hankel determinants generated by x ±1/2 exp(−x), over a 2 < x < ∞. Section 3 is devoted to the computation of the smallest eigenvalue distribution of the LUE and we obtain an asymptotic expansion of the large gap probability, including the hardto-come-by constant term. In section 4, we note the elementary fact that for any polynomials orthogonal with respect to an even weight, a doubling process which folds interval, for example, from (−∞, ∞) to (0, ∞), and (−1, 1) to (0, 1), transforms the problem with two discontinuities, (due to χ (−a,a) (x)) to problems with one gap. This simplifies things considerably. From these one gap problems, combined with the large n asymptotic expansion of the deformed orthogonal polynomials, we compute the constants, that appear in the asymptotic expansion. We investigate the smallest eigenvalue distribution of JUE in section 5, and found the asymptotic expansion for large gap probability, together with the constant term. We present in section 6, the asymptotic gap probability of the (symmetric) Jacobi ensembles where the back ground weight reads (1 − x 2 ) β , |x| < 1.

Gap Probability of the Gaussian Unitary Ensembles
The weight function reads According to (1.1) and the theory of orthogonal polynomials [30], we know that the probability that (−a, a) is free of eigenvalues in the Gaussian unitary ensembles is given by .
Here h j (a) is the square of the L 2 norm of the monic polynomials orthogonal with respect to w(x, a): where P j (x; a) can be normalized (since the weight is even) as [17] P j (x; a) = x j + p(j, a)x j−2 + · · · + P j (0; a).
Remark. The dependence of P n (x; a) on a is seen from its determinant representation in terms of the moments,or the Heine Formula, see [?] eq.(2.2.10).
Remark. It should also be pointed out that P n (z; a) contains only the terms x n−j , j ≤ n and even, since the weight function w(x, a) defined on R is even. This implies that P n (−x; a) = (−1) n P n (x; a), P n (0; a)P n−1 (0; a) = 0.
Remark. On could have written the gap probability as det(I − K ( a, a)) as in per [28], where K is the Christoffel-Darboux kernel which acts on functions as follows: a −a K n (x, y)f (y)dy.
In this setting the kernel is the re-producing kernel constructed out of the "free" or "unperturbed" orthogonal polynomials. For the finite n problem it is the Hermite polynomials. Under "double scaling", see Theorem 2.2, this becomes becomes the sine kernel.

The σ form of Painlevé V
We state here the difference and differential equations satisfied by σ n (a) := a d da log P(a, n) = a d da log D n (a).
Note that P(a, n) = Dn(a) Dn(0) . Theorem 2.1. For a fixed a, σ n satisfies the second order difference equation (2.2) and the following second order fourth degree differential equation: is fixed as a → 0, n → ∞, and let 2n .
We shall be concerned with the behavior of the gap probability for large variable τ. We can of course, make use of (2.4) to investigate the asymptotic behavior of the gap probability. However, we found a convenient way motivated by the relation between Hermite and Laguerre polynomials (with special values of the parameter α) given by Szegö [35] (formula (5.6.1)).
Define the Hankel determinants generated by t − 1 2 e −t and t respectively, we readily see that Before proceeding any further we describe in the next section the smallest eigenvalue distribution of the Laguerre unitary ensembles.

The Smallest Eigenvalue Distribution of the Laguerre Unitary Ensembles
In this section we shall be concerned with the Laguerre weight The probability that all the eigenvalues are greater than t in the finite n Laguerre unitary ensembles, is given by Note that D n (0, α) has a closed-form expression and reads (see [30], p. 321) where G(·) denotes the Barnes-G function, that satisfies the functional relation G(z+1) = Γ(z)G(z), with the 'initial' condition, G(1) = 1. For a comprehensive exposition on the G and other related functions see [38].
Let h j (t, α) be the square of the L 2 norm of monic polynomial P j (.; t, α) orthogonal with respect with the monic P j (x; t, α) reads, It is a well-known fact that the Hankel determinant D n (t, α) can be evaluated as the product of We list here a number of facts about orthogonal polynomials.

The σ from of Painlevé V for finite n and of Painlevé III
Since our Hankel determinant can also be written as , we see that this is a special case, of the Hankel determinant generated by the discontinuous Laguerre [2], by putting A = 0 and B = 1. The parameters A and B here should not be confused with integration interval in page 4.
Remark In [2], the parameters A and B in the weight did not appear in the σ−form of the Painlevé V.
To study the large n behavior of D n (t, α), we first recall some results in [2].
where P n (t; t, α) is the evaluation of the orthogonal polynomial P n (z; t, α) at z = t. Then R n (t) satisfies the following second order differential equation

5)
and can be transformed into a particular Painlevé V [27], namely, The appearance of n in the parameter of the P V indicates that we are studying the finite n problem.
satisfies the following Jimbo-Miwa-Okamoto σ form [27] of Painlevé V equation: Furthermore, H n is expressed in terms of R n by It was pointed out in [2], by changing variable t → s 4n and H n (t) → σ(s) in (3.6), the coefficient of the highest order term in n gives rise to the Jimbo-Miwa-Okamoto σ form of Painlevé III. So we treat R n and equation (3.5), with n large, in a similar way. Here are the results.
then the following second order differential equation holds Then σ(s) satisfies the Jimbo-Miwa-Okamoto σ form of P III (see (3.13) of [26]) The quantity σ(s) when expressed in terms of R(s) reads, Hence, P(s, α) has the following integral representation (3.11) We shall make use of (3.7) to derive the series expansion of R(s) as s → ∞, and then apply (3.11) to obtain P(s, α) for large s. The lemma below gives the bounds for R(s), which we will see is important for later development.
Lemma 3.4. R n (t) and R(s) are bounded by Proof. Noting that To continue, we obtain, neglecting the derivatives in (3.7) and replacing R(s) by R(s), we obtain So it seems reasonable to assume R(s) has an expansion of the form Substituting the above into (3.7), followed by comparing the corresponding coefficients on both sides, we find a 0 = 1 and a 1 = ±α. Since R(s) < 1, we choose a 1 = −α. By direct computations, we eventually arrive at the following expansion formula for R(s).

The evaluation of P n (0; t, α) via Dyson's Coulomb fluid
Recall that the equilibrium density ρ(x) is given by (1.2) with a = t: Here We find with the integral identities in the Apoendix, (3.16) Note that condition √ tb > α and that show that ρ(x) > 0 for t < x < b. Hence, from the normalization condition b t ρ(x)dx = n, it follows that from which we obtain a cubic equation Since the recurrence coefficient α n , is asymptotic to the centre of mass of the support [t, b], namely b+t 2 for large n, see for example [10], we find by using the equality α n = 2n + 1 + α + tR n (see (3.9), [2]) that b ∼2α n − t = 2(2n + 1 + α + tR n ) − t ∼4n + 2α + t + 2t(R n − 1) where the last inequality is due to R n < 1. Now we are in a position to derive the large n expansion for b.
Moreover, putting t = s 4n which tends to 0 as n → ∞, we find (3.20) Proof. Given t > 0, we find from (3.18) that Hence, we assume the following expansion, Substituting this into (3.18), by comparing the corresponding coefficients on both sides, we get , and S 2 (0; t, α) = − n log With the aid of the integral identities listed in the Appendix, and choosing by the branch −t = te πi and −b = be πi , we obtain Finally, by using (3.19), we give an evaluation of P n (0; t, α) for large n and thus determine the constant c 1 (α). Hence, under the assumption that t → 0 and n → ∞ such that s = 4nt is fixed, we find Corollary 3.9. The constant c 1 (α), appearing in (3.14), is identified to be where G(·) is the Barns-G function.
Proof. In what follows, the symbol ∼ refers to 'asymptotic to' for large n.

The Asymptotics of the Gap Probability Distribution of the Gaussian Unitary Ensembles
The gap probability distribution of GUE on (− b √ 2n , b √ 2n ) with n large enough is described by Ehrhardt [21]: where det(I − K 2b ) is the Fredholm determinant with K 2b having the kernel sin(x−y) π(x−y) χ (−b,b) (y). For this reason, we shall make use of the asymptotic expression (3.14) for large n Hankel determinant to deal with our problem under the double scaling Note that τ = 2 √ 2n a.  . From this, we find so that, in view of (3.1) and noting that b 2 ∼ 4k a 2 as k → ∞, Here P(a 2 , α, k) is the probability that all the eigenvalues of k × k Hermitian matrices with weight x α e −x are greater than a 2 and P (b 2 , α) defined by (3.8) is the scaled limiting probability of P(a 2 , α, k), i.e. P (b 2 , α) = lim k→∞ P b 2 4k , α, k . Therefore, according to (3.14), we see that where the constant term reads taking note that G( 3 2 ) = Γ( 1 2 )G( 1 2 ). It follows from eq.(6.39) of Voros [38], that, where ζ(·) is the Riemann zeta function, and lead to The proof is completed.

The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles
Let P(t, α, β, n) denote the probability that all the eigenvalues of the Jacobi unitary ensembles with 1], are between t and 1, where t > 0. It was shown in [16] that satisfies the Jimbo-Miwa-Okamoto σ form of Painlevé VI. Here d 1 and d 2 are constants depending on n, α and β. Under the assumption that t → 0 and n → ∞ such that s = 4n 2 t is fixed, we prove that the σ form of Painlevé VI is reduced to the σ form of Painlevé of a particular III satisfied by s d ds log P( s 4n 2 , α, β, n). Thus, once again we can derive the asymptotic expansion for the doublescaled probability.
Remark It will be shown later that the parameter β does not appear in this Painleve III.
Remark We shall see later that, Remark The constant reads.
whose determination is based on the evaluation, at z = 0 and z = 1, of the monic polynomials P n (z; t, α, β) orthogonal with respect to x α (1 − x) β over the interval [t, 1].
The constant c 2 (α, β) appears to be new.
Hence, in order to carry out large n analysis of P(t, α, β, n) , we first restate the main result in [16]. Here we use the symbol σ n (t) instead of σ(t) for the convenience of later discussion.
Proposition 5.1. The quantity with satisfies the following Jimbo-Miwa-Okamoto [27] σ form of Painlevé VI with parameters and the initial conditions Remark The parameters A and B did not appear in the equation satisfied by the σ−function, [16].
We see that the weight x α (1 − x) β , x ∈ [0, 1], can be converted to the standard Jacobi weight 1]. Furthermore, the following property (see [35], Theorem 8.21.12) holds lim n→∞ n −α P (α,β) where J α (·) is the Bessel function of the first kind of order α. This motivates us to execute the double scaling, namely, t → 0 and n → ∞, such that s = 4n 2 t is fixed. With t = s 4n 2 , we obtain the following result from Proposition 5.1. Then σ(s) satisfies the Jimbo-Miwa-Okamoto σ form of P III (see (3.13) of [26]) with the initial conditions Proof. From (5.1) and (5.7), we find Upon replacing σ n (t) by −σ(s) − s 4 + d 2 and t by s 4n 2 in (5.8), the coefficient of the highest order term in n leads to the desired equation.
Remark The non-appearance of the β parameter in (5.10), does not necessary imply the the non-appearance of β in the asymptotic expansion of P(s, α, β) Before proceeding to the derivation of the expansion σ(s), for large s, we first point out the following important fact about σ(s). Proof. From the initial conditions satisfies by σ n (t): Assuming the continuity of σ n (t) in t, for small t, we see that σ n (t) − d 2 is negative when t is sufficiently enough to 0. Hence, in view of (5.11), we get as t → 0, which completes the proof.

5.2
The evaluation of P n (0; t, α, β) and P n (1; t, α, β) via Dyson's Coulomb gives us the equilibrium density of the fluid, An easy computation shows that By means of the integral identities listed in the Appendix, the normalization condition b t ρ(x)dx = n reads This gives rise to an algebraic equation of degree four in b, from which it follows that We do not display the O(n −3 ) term as it would not affect the outcome.
We follow the methodology in the previous section for LUE. According to (1.3), the monic orthogonal polynomials P n (z; t, α, β) associated with x α (1 − x) β , x ∈ [t, 1], evaluated at z = 0, is given by With the aid of the integral identities in the Appendix, and by taking the branch −t = te πi and Now we discuss the approximation of P n (z; t, α, β) at z = 1. Once again from (1.3), we obtain , and, by applying the integral identities in the Appendix, Finally, pooling together the above results, we give an evaluation of P n (0; t, α, β) and P n (1; t, α, β) as n → ∞.
Proof. Again, in this proof, the symbol ∼ refers to 'asymptotic to' for large n.

Unitary Ensembles
The probability that the interval (−a, a) has no eigenvalues in the (symmetric) Jacobi unitary ensembles with the weight is given by Here w(x, a, β) is the discontinuous Jacobi weight with two jumps where θ(x) is 1 if x ≥ 0 and 0 otherwise. We shall be concerned with the behavior of P(a, β, n) under double scaling.

Conclusion
We obtained in this paper the constant term in the asymptotic expansion of the large gap probability of the Gaussian unitary ensembles; reproducing the Widom-Dyson constant. This is done through the study of the smallest eigenvalue distribution of the Laguerre unitary ensembles, and specializing α = ±1/2, in the constant obtained. Finally, we derive the asymptotic expansion of the smallest eigenvalue distribution of the Jacobi unitary ensembles (x α (1 − x) β , x ∈ (0, 1), α > −1, β > 0). In this situation, although the double scaled σ equation is identical with the LUE σ− equation, however the constant in the JUB problem depends on β.

Appendix: Some Relevant Integral Identities
We list here some integrals, which are relevant to our derivation and can be found in [9], [14] and [23]. The basic assumption is that 0