Computable structural formulas for the distribution of the $\beta$-Jacobi edge eigenvalues

Abstract

The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy’s largest root test in multivariate statistics) and smallest (e.g. condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential–difference system fundamental in the theory of the Selberg integral to provide a recursive scheme to compute the corresponding coefficients.

Appendix

The $\beta$-circular ensemble (1.21) is a particular case of the (generalised) circular Jacobi $\beta$-ensemble. The latter is specified by the family of probability density functions on the unit circle proportional to

$$\begin{aligned} \prod _{l=1}^N \chi _{-\pi< \theta _l< \pi } \,e^{b_2 \theta _l} |1 + e^{i \theta _l} |^{2 b_1} \prod _{1 \le j < k \le N} | e^{i \theta _k} - e^{i \theta _j} |^\beta ; \end{aligned}$$

(A.1)

thus we set $b_1 = b_2 = 0$. Introducing $\xi _l = e^{i \theta _l}$ $(l=1,\dots ,N)$, and temporarily requiring that $b_1$ and $\beta /2$ be positive integer, the measure associated with (A.1) maps to the measure proportional to

$$\begin{aligned} \prod _{l=1}^N \xi _l^{{\tilde{\lambda }}_1} (1 - \xi _l)^{{\tilde{\lambda }}_2} \prod _{1 \le j < k \le N} (\xi _k - \xi _j)^\beta d \xi _1 \cdots d \xi _N \end{aligned}$$

(A.2)

with

$$\begin{aligned} {\tilde{\lambda }}_1 = - b_1 - 1 - (\beta /2) (N-1), \qquad {\tilde{\lambda }}_2 = b_1 + i b_2. \end{aligned}$$

(A.3)

In particular, this tells us that the averages

$$\begin{aligned} \Big \langle \prod _{l=1}^N ( x- e^{i \theta _l})^\alpha {e_p(x-e^{i \theta _1},\dots ,x-e^{i \theta _N}}) \Big \rangle \end{aligned}$$

(A.4)

with respect to (A.1) satisfy the same recurrences (2.2) as the corresponding averages (2.1) for the Jacobi ensemble. We remark that this same conclusion can be reached by direct application of integration by parts as used in [17, 19, §4.6], without the need to assume $b_1$ and $\beta /2$ are positive integers.

We would like to make use of the recurrences satisfied by (A.4) to provide a recursive computational scheme for the circular ensemble gap probability

$$\begin{aligned}&E_N^{\mathrm{C}}(0;(0,\phi );\beta )\nonumber \\&\quad = \lim _{\mu \rightarrow 0} {1 \over {\mathcal {N}}_{\beta ,N}} \int _\phi ^{2 \pi } d \theta _1 \cdots \int _\phi ^{2 \pi } d \theta _N \, \prod _{l=1}^N e^{i \theta _l \mu } \prod _{1 \le j < k \le N} | e^{i \theta _k} - e^{i \theta _j} |^\beta , \end{aligned}$$

(A.5)

where the parameter $\mu$ is introduced for later convenience. As for the Jacobi gap probability $E_N^{\mathrm{J}}(0;(s,1);\lambda _1,\lambda _2,\beta )$ in the parameter range (3), the case of $\beta$ a positive integer is special in this regard. Thus, with the ordering

$$\begin{aligned} R_N: \, \, 2 \pi> \theta _1> \theta _2> \cdots> \theta _N > \phi , \end{aligned}$$

(A.6)

analogous to (1.14) we have

$$\begin{aligned} \prod _{1 \le j< k \le N} | e^{i \theta _k} - e^{i \theta _j} |^\beta = \chi \prod _{l=1}^N e^{-i\theta _l \beta (N-1)/2} \prod _{1 \le j < k \le N} (e^{i \theta _j} - e^{i \theta _k} )^\beta , \end{aligned}$$

(A.7)

where here $\chi$ is a phase factor, $|\chi | = 1$, which has a polynomial structure. In particular for $\beta$ a positive integer, the multidimensional integral is a finite series in powers of $e^{i \phi }$, although taking the limit $\mu \rightarrow 0$ will introduce factors which are polynomials in $\phi$ itself.

From the working of the above paragraph, it suffices to specify a computational scheme for the integrals

$$\begin{aligned} \int _{R_N} d \theta _1 \cdots d \theta _N \, \prod _{l=1}^N e^{i \theta _l {\tilde{\mu }}} \prod _{1 \le j < k \le N} ( e^{i \theta _j} - e^{i \theta _k} )^\beta . \end{aligned}$$

(A.8)

This is done as for the computation of $E_N^{\mathrm{J}}(0;(s,1);\lambda _1,\lambda _2,\beta )$ in the parameter range (3), as detailed in the discussion about (2.27). Actually it is a little simpler, since instead of making repeated use of (1.16), we only require use of

$$\begin{aligned} \int _0^\phi d \theta \, e^{i \theta \nu } = {1 \over i \nu } (e^{i \phi \nu } - 1). \end{aligned}$$

This one-dimensional integral is required for the initial condition $N=1$, and then the evaluation of the case $N=n$ of (A.8) from knowledge of the explicit fractional power series form of

$$\begin{aligned} \int _{R_{n-1}} d \theta _1 \cdots d \theta _{n-1} \, \prod _{l=1}^{n-1} e^{i \theta _l {\tilde{\mu }} } ( e^{i \theta _l} - e^{i \phi } )^\beta \prod _{1 \le j < k \le n-1} ( e^{i \theta _j} - e^{i \theta _k} )^\beta . \end{aligned}$$

(A.9)

This in turn is deduced from knowledge of the same expansion for the case $N=n-1$ of (A.8), then applying the recurrence (2.2) with parameters compatible with (A.3).

For $\beta$ even it is possible to establish a direct relation between $E_N^{\mathrm{J}}(0;(s,1);\lambda _1,\lambda _2,\beta )$ and $E_N^{\mathrm{C}}(0;(0,\phi );\beta )$. First, from the definitions we have

$$\begin{aligned}&E^{\mathrm{J}}_N(0;(s,1);0,\lambda _2,\beta ) = E^{\mathrm{J}}_N(0;(0,1-s);0,\lambda _2,\beta ) \\&\quad = {1 \over J_{N,\lambda _2,0,\beta }} \int _{1-s}^1 dx_1 \cdots \int _{1-s}^1 dx_N \, \prod _{l=1}^N x_l^{\lambda _2} \prod _{1 \le j < k \le N} | x_k - x_j|^\beta . \end{aligned}$$

For $\beta$ even, the expansion (1.14) is valid without any need to order the variables. Substituting in the above gives the formula

$$\begin{aligned}&E^{\mathrm{J}}_N(0;(s,1);0,\lambda _2,\beta ) \nonumber \\&\quad = {1 \over J_{N,\lambda _2,0,\beta }} \sum _\kappa \alpha _\kappa \prod _{l=1}^N {1 \over \kappa _l + \lambda _2 + 1} \Big ( 1 - (1 - s)^{\kappa _l + \lambda _2 + 1} \Big ). \end{aligned}$$

(A.10)

Consider now $E_N^{\mathrm{C}}$. From the definitions,

$$\begin{aligned}&E^{\mathrm{C}}_N(0;( 0,\phi );\beta ) = E^{\mathrm{C}}_N(0;( 2 \pi - \phi , 2 \pi );\beta ) \\&\quad = {1 \over {\mathcal {N}}_{\beta ,N}} \int _0^{2 \pi - \phi } d \theta _1 \cdots \int _0^{2 \pi - \phi } d \theta _N \, \prod _{1 \le j < k \le N} | e^{i \theta _k} - e^{i \theta _j} |^\beta . \end{aligned}$$

For $\beta$ even we can use the expansion (A.7) without the need to impose the ordering (A.6). This shows

$$\begin{aligned}&E^{\mathrm{C}}_N(0;( 0,\phi );\beta ) \nonumber \\&\quad = {{\tilde{\chi }} \over {\mathcal {N}}_{\beta ,N}} \lim _{\mu \rightarrow 0} \sum _\kappa \alpha _\kappa \prod _{l=1}^N {1 \over (\mu - (N-1) \beta /2 + \kappa _l } \Big (1 - e^{- i \phi (\mu - (N-1) \beta /2 + \kappa _l } \Big ),\nonumber \\ \end{aligned}$$

(A.11)

where ${\tilde{\chi }}$ has modulus 1. Comparison of (A.10) and (A.11) shows that for $\beta$ even we can map from $E^{\mathrm{J}}_N$ to $E^{\mathrm{C}}_N$ by setting $\lambda _2 +1= \mu - (N-1) \beta /2$ and $1 - s = e^{- i \phi }$, then taking the limit $\mu \rightarrow 0$ and adjusting the normalisation.