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Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

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Abstract

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi β-ensembles of N×N random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N→∞. In the bulk of the spectrum of each β-ensemble, the same scaling limit is found to be \(e^{p_{1}}{}_{1}F_{1}\), whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre β-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when β is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson’s lemma and the steepest descent method for integrals of Selberg type.

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Notes

  1. Another method for treating the absolute value of integral (1.11) was proposed in [14]. Its first step consists in changing the n contours \(\mathcal{C}\) into a series of n distinct contours, \(\mathcal{C}_{1}\) for variable t 1, \(\mathcal{C}_{2}\) for variable t 2, and so on, so that |Δ(t)|ν can be written as Δ(t)ν without affecting the value of the integral. For instance, if \(\mathcal{C}=\mathbb{R}\), then the contours \(\mathcal{C}_{1},\ldots, \mathcal{C}_{n}\) can be chosen such that they all start at −∞ and comply with ℜ(t n )≤⋯≤ℜ(t 1)≤∞. Although with this method one easily predicts the correct asymptotic expansions, we prefer not to use it because the nature of the contours \(\mathcal{C}_{j}\) depends on the specific integrals that must be evaluated, and because treating rigorously a succession of n parametric integrals is much harder than what we use here.

  2. Following our notation, the polynomial kernel K N (x,y) of the unitary ensembles is given by \(\frac{\varphi _{N}(x)\varphi_{N-1}(y)- \varphi_{N}(y)\varphi_{N-1}(x)}{x-y}\) if n=m=1 and β=2.

  3. By summing up all the n equations in (5.3), one easily shows that the multivariate Airy function is one possible solution of the following (single) PDE: D 0 F=p 1(s)F, where D 0 is the differential operators defined in (2.2). This PDE was first given [13, Eq. (5.17)] as an equation satisfied by the Airy function defined in (1.12).

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Acknowledgements

The work of P.D. was supported by FONDECYT grant #1090034 and by CONICYT through the Anillo de Investigación ACT56. The work of D.-Z.L. was supported by FONDECYT grant #3110108 and partially by the National Natural Science Foundation of China (Grant No. 11171005 and 11301499).

Author information

Authors and Affiliations

  1. Instituto Matemática y Física, Universidad de Talca, 2 Norte 685, Talca, Chile

    Patrick Desrosiers & Dang-Zheng Liu

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China

    Dang-Zheng Liu

Authors
  1. Patrick Desrosiers
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Correspondence to Patrick Desrosiers.

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Communicated by Arno Kuijlaars.

Appendix:  Notation and Constants

Appendix:  Notation and Constants

Most of the constants used in the article can be derived from Selberg’s integrals [27, 50]:

$$\begin{aligned} S_{N}(\lambda_{1},\lambda _{2},\lambda_3) :=& \int_{[0,1]^N}\prod _{i=1}^N x_i^{\lambda _1}(1-x_i)^{\lambda_2} \prod_{1\leq k<j \leq N}|x_k-x_j|^{2\lambda _3} \,d^{N}x \\ =&\prod_{j=0}^{N-1} \frac{\varGamma(1+\lambda_3+j\lambda_3) \varGamma(1+\lambda _{1}+j\lambda_3) \varGamma(1+\lambda_{2}+j\lambda_3)}{ \varGamma(1+\lambda_3)\varGamma(2+\lambda_{1}+\lambda _{2}+(N+j-1)\lambda_3)}. \end{aligned}$$
(A.1)

In particular, one readily shows that

$$ W_{\lambda_{1}, \beta ,N}=(2/\beta)^{(1+\lambda_{1})N+\beta N(N-1)/2}\prod _{j=0}^{N-1} \frac{\varGamma(1+\beta/2+j\beta/2) \varGamma(1+\lambda_{1}+j\beta/2)}{ \varGamma(1+\beta/2)}, $$
(A.2)

and

$$ G_{\beta,N}=\beta^{-N/2-\beta N(N-1)/4}(2 \pi)^{N/2}\prod_{j=0}^{N-1} \frac{\varGamma(1+\beta/2+j\beta/2)}{ \varGamma(1+\beta/2)}. $$
(A.3)

For the Gaussian case, it is often more convenient to use the following integral:

$$ \int_{\mathbb{R}^n}\prod _{i=1}^n e^{-zx_i^2/2}\prod _{1\leq k<j \leq n}|x_k-x_j|^{\beta}\, d^{n}x=\frac{1}{z^{(n+\beta n(n-1)/2)/2}}\varGamma_{\beta,n},\qquad \textrm{Re} \{z\}>0, $$
(A.4)

where

$$ \varGamma_{\beta,n}=(2\pi)^{n/2}\prod _{j=1}^n\frac{\varGamma(1+j\beta /2)}{\varGamma(1+\beta/2)}. $$
(A.5)

The Morris normalization constant is

$$ M_n(a,b, \alpha)= \prod_{j=0}^{n-1} \frac{\varGamma(1+ \alpha+j \alpha) \varGamma(1+a+b+j \alpha)}{ \varGamma(1+ \alpha)\varGamma(1+a+j \alpha)\varGamma(1+b+j \alpha)}. $$
(A.6)

The constants for the soft edge are

$$ \varPhi_{N,n}= \begin{cases} N^{\beta' n(n-1)/12+n/6}\exp\{ -n N(1+\ln2-\ln N-2i\pi)/2\} &\mathrm{H}\beta\mathrm{E},\\ 2^{-\beta' n(n-1)/6-\beta' n/2+2n/3} N^{\beta' n(n-1)/12+\beta ' n \lambda_1/4+ n/6} \\ \quad {}\times \exp\{ -n N(1-\ln N-i\pi)\}&\mathrm{L}\beta\mathrm{E}. \end{cases} $$
(A.7)

In the bulk with n=2m, the constants are

$$ \varPsi_{N,2m}= \begin{cases}(\pi\rho)^{\beta' m(m+1)/2-m} N^{\beta' m^{2}/2}\exp\{ -m N(1+\ln2-\ln N)\} &\mathrm{H}\beta\mathrm{E},\\ (\pi\rho/2)^{\beta' m(m+1)/2-m} N^{\beta' m(m+\lambda _1)/2}\exp\{ -2m N(1-\ln N)\} &\mathrm{L}\beta\mathrm{E},\\ (\pi\rho)^{\beta' m(m+1)/2-m} N^{\beta' m^{2}/2}2^{-\beta' m^{2}/2-\beta' m(\lambda_{1}+\lambda_{2}+1)+2m(1-2N)} & \mathrm{J}\beta\mathrm{E}, \end{cases} $$
(A.8)

where \(\rho=\frac{2}{\pi}\sqrt{1-u^{2}}\), \(\frac{2}{\pi}\sqrt {\frac{1-u}{u}}\), \(\frac{1}{\pi}\frac{1}{\sqrt{u(1-u)}}\), respectively correspond to the HβE, LβE, and JβE.

The constants for the bulk with n=2m−1 are

$$ \varPsi_{N,2m-1}^{(l)}= \begin{cases} \binom{2m-1}{m} \varGamma_{\beta',m-1} \varGamma_{\beta ',m}(\varGamma_{\beta',2m-1})^{-1}(\pi\rho)^{\beta'(m^{2}-1)/2-(2m-1)/2} \\ \quad {}\times N^{\beta' m(m-1)/2} \exp\{-(2m-1) N(1+\ln2-\ln N)/2\} \\ \quad {}\times (\sqrt{N/2})^{-(2m-1)l} (2i\sqrt{\pi\rho/2}) & \mathrm{H}\beta\mathrm{E}, \\ \binom{2m-1}{m} \varGamma_{\beta',m-1} \varGamma_{\beta',m}(\varGamma_{\beta',2m-1})^{-1}(\pi\rho/2)^{\beta'(m^{2}-1)/2- m+1} \\ \quad {}\times \sqrt {2}(2\sqrt{u})^{-\beta'/2+1} N^{\beta'm(m-1)/2+\beta' (2m-1)\lambda_1/4} \\ \quad {}\times \exp\{ -(2m-1) N(1-\ln N-i\pi )\} (-N)^{-(2m-1)l} & \mathrm{L}\beta\mathrm{E}, \\ \binom{2m-1}{m} \varGamma_{\beta',m-1} \varGamma_{\beta',m}(\varGamma_{\beta',2m-1})^{-1}(\pi\rho)^{\beta'(m-1)^{2}/2+(\beta'-2m+1)/2} \\ \quad {}\times \sqrt{2}(2\sqrt{u})^{-\beta'/2+1} N^{\beta' m(m-1)/2} i^{-1}(-1)^{n(N-1)} \\ \quad {}\times 2^{-\beta' m(m+1)/2-\beta' (2m-1)(\lambda_{1}+\lambda_{2})/2+(2m-1)(1-2N+2l)+1} \\ \quad {}\times (1-u)^{nl/2} \sqrt[4]{u} & \mathrm{J}\beta\mathrm{E}. \end{cases} $$
(A.9)

For the hard edge, the constants are

$$ \xi_{N,n}= \begin{cases} \frac{W_{\lambda_1+n, \beta,N}}{W_{\lambda_1, \beta ,N}} &\mathrm{L}\beta\mathrm{E},\\ \frac{S_N(\lambda_1+n,\lambda_2;\beta/2)}{S_N(\lambda _1,\lambda_2;\beta/2)} & \mathrm{J}\beta\mathrm{E}. \end{cases} $$
(A.10)

Finally, the universal coefficients are

$$\begin{aligned} & a_{k}(\beta)= (\beta/2)^{(\beta k+1)k}\bigl(\varGamma (1+\beta/2) \bigr)^{k} \prod_{j=1}^{2k} \frac{(\varGamma(1+2/\beta))^{\beta /2}}{\varGamma(1+\beta j/2) } , \end{aligned}$$
(A.11)
$$\begin{aligned} & b_{k}(\beta)= (\beta/2)^{\beta k(k-1)/2}\bigl(\varGamma (1+\beta/2) \bigr)^{k} \prod_{j=0}^{k-1} \frac{\varGamma(1+\beta j/2)}{\varGamma (1+\beta(k+j)/2)} , \end{aligned}$$
(A.12)

and

$$ \gamma_{m}\bigl(\beta'\bigr)=\binom{2m}{m}\prod _{j=1}^m\frac{\varGamma(1+\beta'j/2)}{\varGamma(1+\beta' (m+j)/2)}. $$
(A.13)

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Desrosiers, P., Liu, DZ. Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles. Constr Approx 39, 273–322 (2014). https://doi.org/10.1007/s00365-013-9206-2

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