see that equation (2.3.27) is equivalent to the third order differential
equation given in Proposition 2.7.

The $\beta=1,4$ analogue of equation (2.3.27), obtained by taking
$\rho^{(Cy)}(s)=r(s)/(1+s^{2})$ in equation (2.3.21), is

$(1+s^{2})^{4}r^{(5)}(s)+10s(1+s^{2})^{3}r^{(4)}(s)-(5\tilde{\alpha}-22)(1+s^{2})^{3}r^{\prime\prime\prime}(s)\\\
+(5\tilde{N}-13)(1+s^{2})^{2}r^{\prime\prime\prime}(s)-8(\tilde{\alpha}-1)s(1+s^{2})^{2}r^{\prime\prime}(s)\\\
+2(7\tilde{N}+1)s(1+s^{2})r^{\prime\prime}(s)+4\tilde{\alpha}^{2}(1+s^{2})^{2}r^{\prime}(s)-2\left[(4\tilde{\alpha}-1)\tilde{N}+1\right](1+s^{2})r^{\prime}(s)\\\
+4(\tilde{N}-1)^{2}r^{\prime}(s)-4\tilde{\alpha}^{2}s(1+s^{2})r(s)+4\tilde{\alpha}(\tilde{N}-1)sr(s)=0,$
(2.3.28)

where $\tilde{\alpha}$ and $\tilde{N}$ are as given in Proposition 2.7. In
comparing equations (2.3.27) and (2.3.28) to their counterparts in Proposition
2.7, we see that degree of each of the coefficients (which alternate between
being even and odd in $s$) has been reduced by $2$.

######  Remark 2.10.

Recall from §1.2.1 that the the Cauchy and circular Jacobi ensembles are
related by the Cayley transformation (1.2.25) and stereographic projection
$\lambda_{i}=\cot(\theta_{i}/2)$ (1.2.26). In particular, the eigenvalue
densities of the two ensembles are related by

$\rho^{(Cy)}(\lambda)=2\sin^{2}(\theta/2)\,\rho^{(cJ)}(e^{\mathrm{i}\theta}).$
(2.3.29)

Equivalently, in the notation $r(s)$ of equation (2.3.25) with
$s=\cot(\theta/2)$,

$r(s)=2\rho^{(cJ)}(e^{\mathrm{i}\theta}).$ (2.3.30)

In the case $\alpha=0$, the circular Jacobi weight
$w^{(cJ)}(e^{\mathrm{i}\theta})$ (1.2.28) is a constant and so according to
equation (2.3.30), $r(s)$ is then also a constant. We can see immediately that
this is consistent with equations (2.3.27) and (2.3.28).

Suppose now $\theta$ is scaled by writing $\theta=2X/N$. Then, in the scaling
limit $N\to\infty$, the circular Jacobi ensemble exhibits a spectrum
singularity at $\theta=0$, in keeping with the discussion in §1.2.1. In view
of equation (2.3.30), in the cases $\beta=1,2$, and $4$, differential
equations for the corresponding density $\rho^{(s.s.)}(X)$ can be obtained by
setting $s=N/X$ and $r(s)=\rho^{(s.s.)}(X)$ in equations (2.3.27) and
(2.3.28), and then equating terms at leading order in $N$. Specifically for
$\beta=2$, we therefore have that $R(X)=\rho^{(s.s.)}(X)$ satisfies the third
order linear differential equation

$X^{2}R^{\prime\prime\prime}(X)+4XR^{\prime\prime}(X)+(2-4\alpha^{2}+4X^{2})R^{\prime}(X)-{4\alpha^{2}\over
X}R(X)=0.$ (2.3.31)

This is consistent with the known exact formula [257], [119, Eq. (7.49) with
$\pi\rho=1$]

$\rho^{(s.s.)}(x)={x\over
2}\Big{(}(J_{\alpha-1/2}(x))^{2}+(J_{\alpha+1/2}(x))^{2}-{2\alpha\over
x}J_{\alpha-1/2}(x)J_{\alpha+1/2}(x)\Big{)},$ (2.3.32)

as can be checked using computer algebra.

Though there is no present benefit in presenting the analogue of Proposition
2.6 for the non-symmetric shifted Jacobi ensemble, there is merit in deriving
the differential equation satisfied by the eigenvalue density of the non-
symmetric CyUE, especially considering its relatively compact form. Repeating
the process outlined above, we first make the change of variables
$x\mapsto(1-x)/2$ in equation (2.3.7) to obtain a differential equation for
the non-symmetric shifted Jacobi unitary ensemble. Setting $a=-N-\alpha$,
$b=-N-\overline{\alpha}$, and replacing $x$ by $\mathrm{i}x$ in this
differential equation, it follows from Proposition 2.5 that we obtain the
differential equation satisfied by $\rho^{(Cy)}(x;N,2)$.

######  Proposition 2.8.

Define the third order differential operator

$\tilde{\mathcal{D}}_{N,2}^{(Cy)}=(1+x^{2})^{3}{\mathrm{d}^{3}\over\mathrm{d}x^{3}}+8x(1+x^{2})^{2}{\mathrm{d}^{2}\over\mathrm{d}x^{2}}\\\
+(1+x^{2})[6+4N(N+\alpha+\overline{\alpha})+(\alpha-\overline{\alpha})^{2}+2\mathrm{i}(\alpha-\overline{\alpha})(2N+\alpha+\overline{\alpha})x+(14-(\alpha+\overline{\alpha})^{2})x^{2}]{\mathrm{d}\over\mathrm{d}x}\\\
+[4+8N(N+\alpha+\overline{\alpha})+3\alpha^{2}-2\alpha\overline{\alpha}+3\bar{\alpha}^{2}+(4-(\alpha+\overline{\alpha})^{2})x^{2}]x\\\
+\mathrm{i}(\alpha-\overline{\alpha})(2N+\alpha+\overline{\alpha})(3x^{2}-1).$
(2.3.33)

For $\beta=2$ and $\alpha\in\mathbb{C}$ with constraint
$\mathrm{Re}(\alpha)>-1/2$, we have

$\tilde{\mathcal{D}}_{N,2}^{(Cy)}\,\rho^{(Cy)}(x)=0.$ (2.3.34)

######  Remark 2.11.

After a simple change of variables, the Jimbo–Miwa–Okamoto $\sigma$-form of
the Painlevé differential equation reads [138, Eq. (1.32)]

$h^{\prime}\Big{(}(1+t^{2})h^{\prime\prime}\Big{)}^{2}+4\Big{(}h^{\prime}(h-th^{\prime})-\mathrm{i}b_{1}b_{2}b_{3}b_{4}\Big{)}^{2}+4\prod_{k=1}^{4}(h^{\prime}+b_{k}^{2})=0.$
(2.3.35)

Let $\mathbf{b}=(b_{1},b_{2},b_{3},b_{4})$ and define
$e_{2}^{\prime}[\mathbf{b}]$, $e_{2}[\mathbf{b}]$ as the elementary symmetric
polynomials of degree two in $\\{b_{1},b_{3},b_{4}\\}$ and
$\\{b_{1},b_{2},b_{3},b_{4}\\}$, respectively. Set

$U_{N}^{(Cy)}(t;(\alpha_{1},\alpha_{2});\xi)=(t^{2}+1){d\over
dt}\log\Big{(}(\mathrm{i}t-1)^{e_{2}^{\prime}[\mathbf{b}]-e_{2}[\mathbf{b}]/2}(\mathrm{i}t+1)^{e_{2}[\mathbf{b}]/2}E_{N}^{(Cy)}((t,\infty);\xi))\Big{)},$
(2.3.36)

where $E_{N}^{(Cy)}$ is specified by equation (2.3.9) in the non-symmetric
case with $\alpha=\alpha_{1}+\mathrm{i}\alpha_{2}$ ($\alpha_{2}\neq 0$). We
have from [138, Prop. 15] that $U_{N}^{(Cy)}$ satisfies the transformed
$\sigma$-PVI equation (2.3.35) with parameters

$\mathbf{b}=(-\alpha_{1},-\mathrm{i}\alpha_{2},N+\alpha_{1},\alpha_{1}).$
(2.3.37)

Analogous to equation (2.3.25), we have from equations (2.3.9), (2.3.36), and
(2.3.37) that

$U_{N}^{(Cy)}(t;(\alpha_{1},\alpha_{2});\xi)=(N+\alpha_{1})\alpha_{2}-\alpha_{1}^{2}t+\xi
r(t)+O(\xi^{2}),\qquad r(t):=(1+t^{2})\rho^{(Cy)}(t).$ (2.3.38)

Substituting (2.3.38) in equation (2.3.35) and equating terms of leading order
in $\xi$, which as for equation (2.3.25) occurs at order $\xi^{2}$, then
differentiating and cancelling a factor of $r^{\prime\prime}$ shows

$(1+t^{2})^{2}r^{\prime\prime\prime}+2t(1+t^{2})r^{\prime\prime}+4[((N+\alpha_{1})\alpha_{2}+\alpha_{1}^{2}t)(r-tr^{\prime})\\\
+((N+\alpha_{1})^{2}-(N+\alpha_{1})\alpha_{2}t-\alpha_{1}^{2}-\alpha_{2}^{2})r^{\prime}]=0.$
(2.3.39)

In the case $\alpha_{2}=0$, this agrees with equation (2.3.27). Now,
substituting for $r(t)$ in terms of $\rho^{(Cy)}(t)$ as specified in equation
(2.3.38) reclaims equation (2.3.34).

####  2.3.3 Differential equations for the Laguerre ensembles

As mentioned at the beginning of this section, the eigenvalue density
$\rho^{(L)}(x)$ of the Laguerre $\beta$ ensemble can be obtained from that of
the Jacobi $\beta$ ensemble via a limiting procedure: From equation (2.0.1),
upon extending the idea of Lemma 1.1, we see (cf. equation (2.1.12))

$\rho^{(L)}(x;N+1,\beta)=\lim_{b\rightarrow\infty}b^{(N+3)N\kappa+(N+2)a+N+1}\,\rho^{(J)}\Big{(}\frac{x}{b};N+1,\beta\Big{)}.$
(2.3.40)

This fact allows one to transform differential equations satisfied by
$\rho^{(J)}(x)$ into analogues satisfied by $\rho^{(L)}(x)$, which will be
presented in a moment.

As an aside, suppose that one would like to obtain differential equations for
$\rho^{(L)}(x)$ without prior knowledge of the analogous differential
equations for $\rho^{(J)}(x)$, i.e., if the results of §2.3.1 were not
available, or if one were interested in ensembles with
$\beta\notin\\{1,2,4\\}$. Then, it is actually more efficient to circumvent
computation of the differential equations for $\rho^{(J)}(x)$ and use the
aforementioned limiting procedure indirectly. That is, let

$L_{\beta,p}^{(N)}(x)=\lim_{b\rightarrow\infty}\left(-\tfrac{x}{b}\right)^{\beta
N+p}J_{\beta,p}^{(N)}(b/x),\qquad\tilde{L}_{\beta,p}^{(N)}(x)=x^{a}e^{-x}\,L_{\beta,p}^{(N)}(x)$

so that by equation (2.0.1), $\rho^{(L)}(x;N+1,\beta)$ is proportional to
$\tilde{L}_{\beta,0}^{(N)}(x)$ when $\beta$ is a positive even integer.
Equation (2.1.17) gives a differential-difference equation for
$\left(-\tfrac{x}{b}\right)^{\beta N+p}J_{\beta,p}^{(N)}(b/x)$ and taking the
limit $b\rightarrow\infty$ thus gives a differential-difference equation for
$L_{\beta,p}^{(N)}(x)$. Substituting
$L_{\beta,p}^{(N)}(x)=x^{-a}e^{x}\tilde{L}_{\beta,p}^{(N)}(x)$ then gives a
differential-difference equation for $\tilde{L}_{\beta,p}^{(N)}(x)$. These
equations with $p=0,1,\ldots,\beta$ are equivalent to matrix differential
equations not unlike (2.3.2) and (2.3.5). However, having taken the limit
$b\rightarrow\infty$, we ensure that these new matrix differential equations
are simpler, and have the added benefit of simplifying down to scalar
differential equations for $\rho^{(L)}(x;N+1,\beta)$ rather than for auxiliary
functions. One may then apply the Stieltjes transform to obtain differential
equations for the resolvents, and use the $\beta\leftrightarrow 4/\beta$
dualities given in Lemma 1.4 to obtain mirror differential equations like
those seen in Theorem 2.2.

Since we are interested in $\rho^{(L)}(x)$ with $\beta\in\\{1,2,4\\}$ and we
have differential equations for $\rho^{(J)}(x)$ for these $\beta$ values, we
use the more direct approach to give the following proposition.

######  Proposition 2.9.

Retaining the definitions of $a_{\beta},N_{\beta}$, and $\tilde{a}$ from
Theorem 2.2, define

$\displaystyle\mathcal{D}_{2,N}^{(L)}$
$\displaystyle=x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+4x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-\left[x^{2}-2(a+2N)x+a^{2}-2\right]x\frac{\mathrm{d}}{\mathrm{d}x}+\left[(a+2N)x-a^{2}\right]$
(2.3.41)

and for $\beta=1$ or $4$,

$\displaystyle\mathcal{D}_{N,\beta}^{(L)}$
$\displaystyle=4x^{5}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+40x^{4}\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}-\left[5\left(\frac{x}{\kappa-1}\right)^{2}-10\left(a_{\beta}+4N_{\beta}\right)\frac{x}{\kappa-1}+5\tilde{a}-88\right]x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}$
$\displaystyle\quad-\left[16\left(\frac{x}{\kappa-1}\right)^{2}-38\left(a_{\beta}+4N_{\beta}\right)\frac{x}{\kappa-1}+22\tilde{a}-16\right]x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$
$\displaystyle\quad+\left[\left(\frac{x}{\kappa-1}\right)^{2}-4\left(a_{\beta}+4N_{\beta}\right)\left(\frac{x}{\kappa-1}\right)+2\left(2\left(a_{\beta}+4N_{\beta}\right)^{2}+\tilde{a}-2\right)\right]\frac{x^{3}}{(\kappa-1)^{2}}\frac{\mathrm{d}}{\mathrm{d}x}$
$\displaystyle\quad-\left[4(\tilde{a}-3)\left(a_{\beta}+4N_{\beta}\right)\frac{x}{\kappa-1}-\tilde{a}^{2}+14\tilde{a}+16\right]x\frac{\mathrm{d}}{\mathrm{d}x}-\left(a_{\beta}+4N_{\beta}\right)\left(\frac{x}{\kappa-1}\right)^{3}$
$\displaystyle\quad+\left(2\left(a_{\beta}+4N_{\beta}\right)^{2}+\tilde{a}\right)\left(\frac{x}{\kappa-1}\right)^{2}-(3\tilde{a}+4)\left(a_{\beta}+4N_{\beta}\right)\frac{x}{\kappa-1}+\tilde{a}^{2}.$
(2.3.42)

Then, for $\beta=1,2$, and $4$,

$\mathcal{D}_{N,\beta}^{(L)}\,\rho^{(L)}(x;N,\beta)=0$ (2.3.43)

and

$\mathcal{D}_{N,\beta}^{(L)}\,\frac{1}{N}W_{1}^{(L)}(x;N,\beta)=\begin{cases}(x+a),&\beta=2,\\\
\frac{4}{\kappa-1}\left[2\left(\frac{x}{\kappa-1}\right)^{2}+(2a_{\beta}-1)\frac{x}{\kappa-1}\right]N_{\beta}&\\\
\quad-\frac{1}{\kappa-1}\left[\left(\frac{x}{\kappa-1}\right)^{3}-(a_{\beta}+2)\left(\frac{x}{\kappa-1}\right)^{2}\right]&\\\
\quad+\frac{1}{\kappa-1}\left[(a_{\beta}^{2}+4a_{\beta}-4)\frac{x}{\kappa-1}-a_{\beta}(a_{\beta}-2)^{2}\right],&\beta=1,4.\end{cases}$
(2.3.44)

###### Proof.

To obtain equation (2.3.43) from (2.3.7) and (2.3.13), change variables
$x\mapsto x/b$, multiply both sides by $b^{(N+2)(N-1)\kappa+(N+1)a+N}$
according to equation (2.3.40), and then take the limit $b\rightarrow\infty$.
This is equivalent to changing variables $x\mapsto x/b$, extracting terms of
leading order in $b$, then rewriting $\rho^{(J)}$ as $\rho^{(L)}$.

To obtain equation (2.3.44), apply the same prescription to equations (2.3.8)
and (2.3.14). Alternatively, one may apply the Stieltjes transform to equation
(2.3.43). This is considerably easier than in the Jacobi case (see Appendix B)
since

$\displaystyle\int_{0}^{\infty}\frac{x^{n}}{s-x}\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\rho^{(L)}(x)\,\mathrm{d}x=s^{n}\frac{\mathrm{d}^{n}}{\mathrm{d}s^{n}}W_{1}^{(L)}(s)$

can be computed through repeated integration by parts using the fact that the
boundary terms vanish at all stages: For $a>-1$ real and $m>n$ non-negative
integers,
$\frac{x^{m}}{s-x}\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\rho^{(L)}(x)$ has a
factor of $x^{a+m-n}$ which dominates at $x=0$ and a factor of $e^{-x}$ which
dominates as $x\rightarrow\infty$. ∎

Equation (2.3.43) with $\beta=2$ is equivalent to that seen in [164], [2].
When $\beta=1,4$, it has been checked for $N=1,2$ using computer algebra. From
inspection, it seems that the natural variable of equations (2.3.43) and
(2.3.44) is $x/(\kappa-1)$, which is the limit
$\lim_{b\rightarrow\infty}b_{\beta}\left(\tfrac{x}{b}\right)$. This is in
keeping with the duality relation on the resolvent presented in Lemma 1.4.
Strictly speaking, changing variables to $y=x/(\kappa-1)$ is not natural and
would be counterproductive since the corresponding weight
$\left[(\kappa-1)y\right]^{a}e^{(1-\kappa)y}$ vanishes at $\kappa=1$ and has
different support depending on whether $\kappa<1$ or $\kappa>1$.

####  2.3.4 Differential equations for the Gaussian ensembles

The previous subsection contains a discussion on how one would obtain
differential equations for the densities and resolvents of Laguerre ensembles
with even $\beta$ without knowledge of differential equations for the
corresponding Jacobi $\beta$ ensembles’ densities and resolvents. We now
elucidate those ideas by explicitly applying them to the study of the Gaussian
ensembles with $\beta=6$ and consequently, by duality, $\beta=2/3$. Indeed,
since we do not have differential equations for the $\beta=6$ Jacobi ensemble
at hand, we cannot immediately apply the direct limiting approach used in the
proof of Proposition 2.9. Thus, for our purposes, it is in fact more efficient
to replicate the proofs of §2.3.1 while taking limits at an earlier stage so
as to circumvent the need for investigating the $\beta=6$ Jacobi ensemble
altogether.

Like in the Jacobi and Laguerre cases, our initial focus is the average

$I_{N,\beta}^{(G)}(\lambda):=\left<\prod_{i=1}^{N}|\lambda-\lambda_{i}|^{\beta}\right>_{\text{GE}_{N,\beta}},$
(2.3.45)

where the subscript $\text{GE}_{N,\beta}$ indicates that the average is with
respect to the eigenvalue j.p.d.f. (1.2.7) with weight
$w^{(G)}(\lambda)=e^{-\lambda^{2}}$. The Gaussian analogue of the duality
relation (2.1.21) is [27]

$\left<\prod_{i=1}^{N}\left(\lambda-\kappa^{-1/2}\lambda_{i}\right)^{n}\right>_{\text{GE}_{N,\beta}}=\left<\prod_{i=1}^{n}(\lambda-\mathrm{i}\lambda_{i})^{N}\right>_{\text{GE}_{n,4/\beta}}.$
(2.3.46)

Replacing $\lambda$ by $\kappa^{-1/2}\lambda$ in the above duality relation
and then factoring $(-\mathrm{i})^{nN}$ from the right-hand side shows that
for even $\beta$, $I_{N,\beta}^{(G)}(\lambda)$ is proportional to
$G_{\beta,0}^{(N)}(\lambda)$, where

$G_{n,p}^{(N)}(\lambda):=(-\mathrm{i})^{nN+p}\left<\prod_{i=1}^{n}\left(\lambda_{i}+\mathrm{i}\kappa^{-1/2}\lambda\right)^{N+\chi_{i\leqslant
p}}\right>_{\text{GE}_{n,4/\beta}},\quad 0\leqslant p\leqslant n.$ (2.3.47)

Setting $a^{\prime}=b^{\prime}=L$ and changing variables
$\lambda_{i}\mapsto\tfrac{1}{2}\left(1+\tfrac{\lambda_{i}}{\sqrt{L}}\right)$
in $J_{n,p}^{(N)}(\lambda)$ (2.1.16), we see that

$G_{n,p}^{(N)}(\lambda)=\lim_{L\rightarrow\infty}4^{nL}(-2\mathrm{i}\sqrt{L})^{nN+p}(2\sqrt{L})^{n(n-1)/\kappa+n}\left.J_{n,p}^{(N)}\left(\tfrac{1}{2}\left(1-\mathrm{i}\sqrt{\tfrac{2}{\beta
L}}\lambda\right)\right)\right|_{a^{\prime}=b^{\prime}=L}.$ (2.3.48)

Thus, equation (2.1.17) simplifies to a differential-difference equation for
$G_{n,p}^{(N)}(\lambda)$,

$(n-p)G_{n,p+1}^{(N)}(\lambda)=\tfrac{(n-p)}{\sqrt{\kappa}}\lambda
G_{n,p}^{(N)}(\lambda)-\tfrac{\sqrt{\kappa}}{2}\frac{\mathrm{d}}{\mathrm{d}\lambda}G_{n,p}^{(N)}(x)\\\
+\tfrac{p}{2}\left(\tfrac{1}{\kappa}(n-p)+N+1\right)G_{n,p-1}^{(N)}(\lambda)$
(2.3.49)

(cf. [133, Eq. (5.5)]). Taking the $L\rightarrow\infty$ limit early has
already yielded a simpler equation than (2.1.17). However, we can take this
one step further by defining

$\tilde{G}_{n,p}^{(N)}(\lambda)=e^{-\lambda^{2}}G_{n,p}^{(N)}(\lambda)$
(2.3.50)

so that $\rho^{(G)}(\lambda;N+1,\beta)$ is proportional to
$\tilde{G}_{\beta,0}^{(N)}(\lambda)$. It is then easy to obtain the
differential-difference equation

$\frac{\mathrm{d}}{\mathrm{d}\lambda}\tilde{G}_{n,p}^{(N)}(\lambda)=\tfrac{p}{\sqrt{\kappa}}\left(\tfrac{1}{\kappa}(n-p)+N+1\right)\tilde{G}_{n,p-1}^{(N)}(\lambda)\\\
+2\left(\tfrac{1}{\kappa}(n-p)-1\right)\lambda\tilde{G}_{n,p}^{(N)}(\lambda)-\tfrac{2}{\sqrt{\kappa}}(n-p)\tilde{G}_{n,p+1}^{(N)}(\lambda).$
(2.3.51)

With $n=\beta\in 2\mathbb{N}$ and $p=0,1,\ldots,n$, this is equivalent to a
matrix differential equation on the vector
$\mathbf{v}=\left[\tilde{G}_{n,0}^{(N)}(x)\,\cdots\,\tilde{G}_{n,n}^{(N)}(x)\right]^{T}$,
which is moreover equivalent to a scalar differential equation for
$\rho^{(G)}(\lambda;N+1,\beta)$.

######  Proposition 2.10.

For $\beta=2/3$ or $6$, define

$\displaystyle\mathcal{D}_{N,\beta}^{(G)}$
$\displaystyle=81(\kappa-1)^{7/2}\frac{\mathrm{d}^{7}}{\mathrm{d}x^{7}}+1008\left(3N_{\beta}-\frac{2x^{2}}{\kappa-1}+2\right)(\kappa-1)^{5/2}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}$
$\displaystyle\quad+2016x(\kappa-1)^{3/2}\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}$
$\displaystyle\quad+64\left(21N_{\beta}-\frac{14x^{2}}{\kappa-1}+5\right)\left(21N_{\beta}-\frac{14x^{2}}{\kappa-1}+23\right)(\kappa-1)^{3/2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}$
$\displaystyle\quad+9984\left(3N_{\beta}-\frac{2x^{2}}{\kappa-1}+2\right)x(\kappa-1)^{1/2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}$
$\displaystyle\quad+256\Bigg{[}54N_{\beta}\left(4N_{\beta}^{2}+8N_{\beta}+3\right)-(432N_{\beta}^{2}+576N_{\beta}+57)\frac{x^{2}}{\kappa-1}$
$\displaystyle\quad+96(3N_{\beta}+2)\frac{x^{4}}{(\kappa-1)^{2}}-\frac{64x^{6}}{(\kappa-1)^{3}}-20\Bigg{]}(\kappa-1)^{1/2}\frac{\mathrm{d}}{\mathrm{d}x}$
$\displaystyle\quad+256\left(144N_{\beta}^{2}+192N_{\beta}-64(3N_{\beta}+2)\frac{x^{2}}{\kappa-1}+\frac{64x^{4}}{(\kappa-1)^{2}}+25\right)\frac{x}{(\kappa-1)^{1/2}},$
(2.3.52)

where we retain the definition $N_{\beta}=(\kappa-1)N$. Then, for these same
$\beta$ values,

$\mathcal{D}_{N,\beta}^{(G)}\,\rho^{(G)}(x;N,\beta)=0$ (2.3.53)

and

$\displaystyle\mathcal{D}_{N,\beta}^{(G)}\,\frac{1}{N}W_{1}^{(G)}(x;N,\beta)$
$\displaystyle=\frac{2^{11}}{\sqrt{\kappa-1}}\left(\frac{4x^{2}}{\kappa-1}-6N_{\beta}-7\right)^{2}-3\frac{2^{12}}{\sqrt{\kappa-1}}.$
(2.3.54)

###### Proof.

Take equation (2.3.51) with $n=\beta=6$ and $N$ replaced by $N-1$ to obtain
the matrix differential equation

$\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}x}=\begin{bmatrix}2x&-4\sqrt{3}&0&0&0&0&0\\\
\frac{3N+5}{3\sqrt{3}}&\frac{4x}{3}&-\frac{10}{\sqrt{3}}&0&0&0&0\\\
0&\frac{6N+8}{3\sqrt{3}}&\frac{2x}{3}&-\frac{8}{\sqrt{3}}&0&0&0\\\
0&0&\sqrt{3}(N+1)&0&-2\sqrt{3}&0&0\\\
0&0&0&\frac{12N+8}{3\sqrt{3}}&-\frac{2x}{3}&-\frac{4}{\sqrt{3}}&0\\\
0&0&0&0&\frac{15N+5}{3\sqrt{3}}&-\frac{4x}{3}&-\frac{2}{\sqrt{3}}\\\
0&0&0&0&0&2\sqrt{3}N&-2x\end{bmatrix}\mathbf{v}$ (2.3.55)

for
$\mathbf{v}=\left[\tilde{G}_{6,0}^{(N-1)}(x)\,\cdots\,\tilde{G}_{6,6}^{(N-1)}(x)\right]^{T}$.
Like in the proofs of Lemmas 2.1 and 2.2, for $1\leqslant p\leqslant 6$, the
$p\textsuperscript{th}$ row of the above matrix differential equation gives an
expression for $\tilde{G}_{6,p}^{(N-1)}(x)$ in terms of
$\frac{\mathrm{d}}{\mathrm{d}x}\tilde{G}_{6,p-1}^{(N-1)}(x)$ and
$\tilde{G}_{6,k}^{(N-1)}(x)$ with $k<p$. Substituting these expressions into
the equation corresponding to the last row in the order of decreasing $p$ then
yields a seventh-order differential equation satisfied by
$\tilde{G}_{6,0}^{(N-1)}(x)$. Since $\rho^{(G)}(x;N,6)$ is proportional to
$\tilde{G}_{6,0}^{(N-1)}(x)$, this equation is equivalent to (2.3.53) for
$\beta=6$. Taking the Stieltjes transform of this result and substituting in
the spectral moments $m_{2}^{(G)}$ and $m_{4}^{(G)}$ from [319] then yields
equation (2.3.54) for $\beta=6$. Employing the duality relation given in Lemma
1.4 for the resolvent then shows that equation (2.3.54) also holds for
$\beta=2/3$. Finally, taking the inverse Stieltjes transform of this result
shows that equation (2.3.53) holds for $\beta=2/3$ as well. ∎

Equation (2.3.53) has been checked for $N=1,2$ using computer algebra. Similar
to the Laguerre case, it seems like $x/\sqrt{\kappa-1}$ is the natural
variable in Proposition 2.10. This is evidently due to the duality relation
used in the proof of this proposition. Like in the Laguerre case, there is
presently no benefit in changing variables to $x/\sqrt{\kappa-1}$.

It has been mentioned that equation (2.3.51) leads to a matrix differential
equation which is equivalent to a scalar differential equation for
$\rho^{(G)}(x;N+1,\beta)$ when $\beta$ is even. For $\beta=2$ and $4$, these
differential equations are respectively

$\frac{\mathrm{d}}{\mathrm{d}x}\begin{bmatrix}\tilde{G}_{2,0}^{(N-1)}(x)\\\
\tilde{G}_{2,1}^{(N-1)}(x)\\\
\tilde{G}_{2,2}^{(N-1)}(x)\end{bmatrix}=\begin{bmatrix}2x&-4&0\\\ N+1&0&-2\\\
0&2N&-2x\end{bmatrix}\begin{bmatrix}\tilde{G}_{2,0}^{(N-1)}(x)\\\
\tilde{G}_{2,1}^{(N-1)}(x)\\\ \tilde{G}_{2,2}^{(N-1)}(x)\end{bmatrix}$
(2.3.56)

and

$\frac{\mathrm{d}}{\mathrm{d}x}\begin{bmatrix}\tilde{G}_{4,0}^{(N-1)}(x)\\\
\tilde{G}_{4,1}^{(N-1)}(x)\\\ \tilde{G}_{4,2}^{(N-1)}(x)\\\
\tilde{G}_{4,3}^{(N-1)}(x)\\\
\tilde{G}_{4,4}^{(N-1)}(x)\end{bmatrix}=\begin{bmatrix}2x&-4\sqrt{2}&0&0&0\\\
\frac{2N+3}{2\sqrt{2}}&x&-3\sqrt{2}&0&0\\\ 0&\sqrt{2}(N+1)&0&-2\sqrt{2}&0\\\
0&0&\frac{6N+3}{2\sqrt{2}}&-x&-\sqrt{2}\\\
0&0&0&2\sqrt{2}N&-2x\end{bmatrix}\begin{bmatrix}\tilde{G}_{4,0}^{(N-1)}(x)\\\
\tilde{G}_{4,1}^{(N-1)}(x)\\\ \tilde{G}_{4,2}^{(N-1)}(x)\\\
\tilde{G}_{4,3}^{(N-1)}(x)\\\ \tilde{G}_{4,4}^{(N-1)}(x)\end{bmatrix}.$
(2.3.57)

The corresponding scalar differential equations for the respective eigenvalue
densities agree with those of Proposition 2.1 after scaling
$x\mapsto\sqrt{\frac{N\kappa}{2g}}x$. So too does the differential equation
for $\rho^{(G)}(x;N,1)$ obtained through the $\beta\leftrightarrow 4/\beta$
duality given in Lemma 1.4.

###  2.4 Scalings of the Differential Equations

The differential equations of the previous section hold true not only for
finite integers $N$, but also in the large $N$ limit, upon appropriate
scaling. In this section we consider two scaling regimes: In §2.4.1, we study
the classical matrix ensembles (and also the Gaussian $\beta$ ensembles with
$\beta=2/3,6$) under the global scalings introduced in Definition 1.6, while
in §2.4.2, we look at the soft and hard edge scaling regimes (recall the
discussion following Proposition 1.2).

In the global scaling regime, we use Theorem 1.2 to expand the scaled
resolvents $\tilde{W}_{1}(x)$, specified by equation (1.1.24), in $1/N$ and
derive first order differential equations which characterise the expansion
coefficients. Being first order, these differential equations are
straightforward to solve, and using the Sokhotski–Plemelj inversion formula
(1.1.25), we are able to compute expressions for the large $N$ limiting forms
$\rho^{0}(\lambda)$ of the smoothed eigenvalue densities and their $1/N$
corrections $\rho^{1}(\lambda)$. As expected, we recover the Wigner semi-
circle, Marčenko–Pastur, and Wachter laws seen in Proposition 1.2, along with
the expression for $\rho^{(Cy),0}(\lambda)$ presented therein. Moreover, in
keeping with Note 1.1 following this proposition, we observe that while the
large $N$ limiting forms $\rho^{0}(\lambda)$ are $\beta$-independent, the
$1/N$ corrections $\rho^{1}(\lambda)$ are different for each $\beta$ that we
consider. This is in keeping with the general-$\beta$ results known from loop
equation analysis (see, e.g., [319], [142] and references therein).

In contrast to the $\beta$-universality seen for the large $N$ limiting forms
of the global scaled eigenvalue densities, we see a different type of
universality at the soft and hard edges. Namely, it is known [115], [256] that
for fixed $\beta$, a soft edge of a classical $\beta$ ensemble behaves
statistically as that of any other classical $\beta$ ensemble, in the large
$N$ limit — likewise for a hard edge. Thus, we may denote the large $N$
limiting forms of the soft and hard edge scaled eigenvalue densities of the
classical $\beta$ ensembles as $\rho^{(soft)}(\lambda;\beta)$ and
$\rho^{(hard)}(\lambda;\beta)$, respectively. In §2.4.2, we apply these
scalings to the differential equations of Section 2.3 to derive linear
differential equations satisfied by $\rho^{(soft)}(\lambda;\beta)$ for
$\beta\in\\{2/3,1,2,4,6\\}$ and $\rho^{(hard)}(\lambda;\beta)$ for
$\beta\in\\{1,2,4\\}$.

####  2.4.1 Global scaled differential equations

Global scaling of the eigenvalue density refers to a choice of scaling which
ensures that, in the large $N$ limit, the scaled density is supported on a
finite interval on which it integrates to one. Such a transformation is not
unique, due to the length of the interval of support being unspecified; for
convenience, we follow the conventions outlined in Definition 1.6, which are
chosen to be consistent with the earlier works [319], [142]. We recall then
that the global scaled eigenvalue densities of the classical $\beta$ ensembles
are

$\displaystyle\tilde{\rho}^{(G)}(\lambda)$
$\displaystyle=\sqrt{\frac{\kappa}{N}}\,\rho^{(G)}(\sqrt{\kappa N}\lambda),$
(2.4.1) $\displaystyle\tilde{\rho}^{(L)}(\lambda)$
$\displaystyle=\kappa\,\rho^{(L)}(\kappa N\lambda),$ (2.4.2)
$\displaystyle\tilde{\rho}^{(J)}(\lambda)$
$\displaystyle=\frac{1}{N}\,\rho^{(J)}(\lambda),$ (2.4.3)
$\displaystyle\tilde{\rho}^{(Cy)}(\lambda)$
$\displaystyle=\frac{1}{N}\,\rho^{(Cy)}(\lambda)\Big{|}_{\alpha=\hat{\alpha}\kappa
N},$ (2.4.4)

with corresponding scaled resolvents (1.1.24),

$\displaystyle\tilde{W}_{1}^{(G)}(x)$ $\displaystyle:=\sqrt{\kappa
N}\,W_{1}^{(G)}(\sqrt{\kappa N}x),$ (2.4.5)
$\displaystyle\tilde{W}_{1}^{(L)}(x)$ $\displaystyle:=\kappa
N\,W_{1}^{(L)}(\kappa Nx),$ (2.4.6) $\displaystyle\tilde{W}_{1}^{(Cy)}(x)$
$\displaystyle:=W_{1}^{(Cy)}(x)\Big{|}_{\alpha=\hat{\alpha}\kappa N},$ (2.4.7)

where $\hat{\alpha}$ is constant in $N$; since the spectrum of the Jacobi
$\beta$ ensemble is already supported on $[0,1]$, there is no need to scale
its resolvent. In §1.1.1, it was mentioned that the scaled resolvents are more
robust than the global scaled eigenvalue densities in the sense that they
admit well-defined large $N$ expansions due to Theorem 1.2. Thus, following
[319], [142] again for consistency, we consider the expansions

$\displaystyle\tilde{W}_{1}^{(G)}(x;N)$
$\displaystyle=2N\sum_{l=0}^{\infty}\frac{W_{1}^{(G),l}(x)}{(2\sqrt{\kappa}N)^{l}},$
(2.4.8) $\displaystyle\tilde{W}_{1}^{(L)}(x;N)$
$\displaystyle=N\sum_{l=0}^{\infty}\frac{W_{1}^{(L),l}(x)}{(\sqrt{\kappa}N)^{l}},$
(2.4.9) $\displaystyle W_{1}^{(J)}(x;N)$
$\displaystyle=N\sum_{l=0}^{\infty}\frac{W_{1}^{(J),l}(x)}{(\kappa N)^{l}},$
(2.4.10) $\displaystyle\tilde{W}_{1}^{(Cy)}(x;N)$
$\displaystyle=N\sum_{l=0}^{\infty}\frac{W_{1}^{(Cy),l}(x)}{(\kappa N)^{l}}.$
(2.4.11)

Substituting these expansions into the differential equations of Section 2.3
(after scaling according to (2.4.5)–(2.4.7)) and equating terms of like order
in $N$ results in linear differential equations for the expansion coefficients
$W_{1}^{l}(x)$. The differential equations for $W_{1}^{l}(x)$ are first order
with inhomogeneous terms dependent on $\\{W_{1}^{k}(x)\\}_{k=0}^{l-1}$. Hence,
they constitute a recursive process for computing the scaled resolvents
$\tilde{W}_{1}(x)$ up to any desired order in $1/N$. The topological recursion
or, equivalently, loop equation formalism [110] (see also the review given in
Section 4.1) accomplishes the same task for general $\beta>0$, but our
approach, valid for the specific $\beta$ values considered, is more efficient
since there is no need to consider the $n$-point correlators $U_{n},W_{n}$ for
$n>1$. As a consequence, our work is able to isolate extra structures, such as
those studied by Haagerup and Thorbjørnsen [170]: In [170], the authors showed
in the GUE case that $W_{1}^{(G),l}(x)$ is a polynomial in $(x^{2}-2)^{-1/2}$,
with the polynomial coefficients satisfying a certain three-term recurrence.

##### Differential equations for $W_{1}^{(G),l}(x)$

We begin with the Gaussian $\beta$ ensembles with $\beta\in\\{2/3,1,4,6\\}$
and refer to [170] for the $\beta=2$ case.

######  Proposition 2.11.

We reuse the notation of Proposition 2.1 with $g=1/2$ so that
$h=\sqrt{\kappa}-1/\sqrt{\kappa}$ and $y_{(G)}:=\sqrt{x^{2}-2}$. Then, for
$\beta=1$ and $4$, the expansion coefficients of
$\tilde{W}_{1}^{(G)}(x;N,\beta)$ (2.4.8) satisfy the differential equations

$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),0}(x)-xW_{1}^{(G),0}(x)=-1,$
(2.4.12)
$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),1}(x)-xW_{1}^{(G),1}(x)=4h\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),0}(x)-\frac{h}{y_{(G)}^{2}}\left[2xW_{1}^{(G),0}(x)+5\right],$
(2.4.13)

and for $l\geqslant 2$, the general differential equation

$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),l}(x)-xW_{1}^{(G),l}(x)=4h\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),l-1}(x)-\frac{2hx}{y_{(G)}^{2}}W_{1}^{(G),l-1}(x)\\\
+\frac{1}{y_{(G)}^{2}}\left[\frac{5y_{(G)}^{2}}{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}-3x\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\frac{\mathrm{d}}{\mathrm{d}x}\right]W_{1}^{(G),l-2}(x)\\\
-\frac{5h}{y_{(G)}^{2}}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(G),l-3}(x)+\frac{1}{y_{(G)}^{2}}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}W_{1}^{(G),l-4}(x),$
(2.4.14)

where we set $W_{1}^{(G),k}:=0$ for $k<0$.

###### Proof.

Beginning with the $\beta=1,4$ case of equation (2.0.4), set $g=1/2$ and map
$x\mapsto\sqrt{\kappa N}x$ to obtain a differential equation for
$\tilde{W}_{1}^{(G)}(x;N,\beta)$. Substitute in the expansion (2.4.8) and
equate terms of like order in $N$. The terms of leading order in $N$
correspond to equation (2.4.12), the next to leading order terms correspond to
equation (2.4.13), and so on. ∎

Let us recall from §1.1.1 the relationship between the resolvent expansion
coefficients $\\{W_{1}^{l}(x)\\}_{l=0}^{\infty}$ and the coefficients
$\\{\tilde{M}_{k,l}\\}_{l=0}^{\infty}$ of the $1/N$ expansions of the spectral
moments $\tilde{m}_{k}$ of $\tilde{\rho}(\lambda)$. The facts that
$\tilde{m}_{0}=m_{0}/N=1$ and $W_{1}^{l}(x)$ has asymptotic expansion
$\sum_{k=0}^{\infty}\tilde{M}_{k,l}/x^{k+1}$ tell us that the solutions of the
differential equations (2.4.12)–(2.4.14) are fully determined by the boundary
conditions at large $x$,

$\displaystyle W_{1}^{0}(x)$ $\displaystyle=\frac{1}{x}+{\rm
O}\left(\frac{1}{x^{2}}\right),$ (2.4.15) $\displaystyle W_{1}^{l}(x)$
$\displaystyle={\rm O}\left(\frac{1}{x^{2}}\right),\quad l>1.$ (2.4.16)

This is in fact true for all of the first order differential equations given
in this subsection.

Now, applying the above proof to differential equation (2.3.54) gives
analogous differential equations for the expansion coefficients
$W_{1}^{(G),l}(x)$ when $\beta=2/3,6$.

######  Proposition 2.12.

Retain the definition of $y_{(G)}$ from Proposition 2.11. Then, for
$\beta=2/3$ and $6$, the expansion coefficients of
$\tilde{W}_{1}^{(G)}(x;N,\beta)$ satisfy the differential equations

$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),0}(x)-xW_{1}^{(G),0}(x)=-1,$
(2.4.17)
$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),1}(x)-xW_{1}^{(G),1}(x)=6h\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),0}(x)-\frac{h}{y_{(G)}^{2}}\left[4xW_{1}^{(G),0}(x)-7\right],$
(2.4.18)

$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),2}(x)-xW_{1}^{(G),2}(x)=6h\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),1}(x)-\frac{4hx}{y_{(G)}^{2}}W_{1}^{(G),1}(x)-\frac{43}{3y_{(G)}^{4}}\\\
+\frac{1}{12y_{(G)}^{4}}\left[49y_{(G)}^{4}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}-78xy_{(G)}^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+3(72-19x^{2})\frac{\mathrm{d}}{\mathrm{d}x}+25x\right]W_{1}^{(G),0}(x),$
(2.4.19)

and for $l\geqslant 3$, the general differential equation

$y_{(G)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),l}(x)-xW_{1}^{(G),l}(x)=6h\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(G),l-1}(x)-\frac{4hx}{y_{(G)}^{2}}W_{1}^{(G),l-1}(x)\\\
+\frac{1}{12y_{(G)}^{4}}\left[49y_{(G)}^{4}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}-78xy_{(G)}^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+3(72-19x^{2})\frac{\mathrm{d}}{\mathrm{d}x}+25x\right]W_{1}^{(G),l-2}(x)\\\
+\frac{h}{3y_{(G)}^{4}}\left[49y_{(G)}^{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+39x\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-10\frac{\mathrm{d}}{\mathrm{d}x}\right]W_{1}^{(G),l-3}+\frac{7h}{y_{(G)}^{4}}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}W_{1}^{(G),l-5}\\\
+\frac{1}{18y_{(G)}^{4}}\left[63y_{(G)}^{2}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+63x\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}+230\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}\right]W_{1}^{(G),l-4}+\frac{3h}{4y_{(G)}^{4}}\frac{\mathrm{d}^{7}}{\mathrm{d}x^{7}}W_{1}^{(G),l-6},$
(2.4.20)

where we again set $W_{1}^{(G),k}:=0$ for $k<0$.

This proposition has been checked against [319] up to $l=6$, and thus also
serves as a check for differential equation (2.3.54) up to order six in $1/N$.
Similar checks for consistency, against [142], have been carried out for
Propositions 2.13 to 2.15 below.

##### Differential equations for $W_{1}^{(L),l}(x)$ and $W_{1}^{(J),l}(x)$

Since the differential equations for the Laguerre and Jacobi ensembles’
resolvents involve the parameters $a,b$, equating terms of equal order in $N$
requires us to choose how these parameters depend on $N$. To make our results
suitable for most known or potential applications [33], [56], [305], [266],
[229], [240], [241], [70], [71] and to improve clarity, we take
$a=\hat{a}\kappa N$ and $b=\hat{b}\kappa N$ with $\hat{a},\hat{b}$ constant in
$N$. Note that our method easily accommodates for more general $N$-expansions
of $a$ and $b$.

######  Proposition 2.13.

Set $a=\hat{a}\kappa N$ with $\hat{a}$ constant in $N$, and let
$y_{(L)}:=\sqrt{(\hat{a}-x)^{2}-4x}$. The expansion coefficients of the LUE
scaled resolvent $\tilde{W}_{1}^{(L)}(x;N,2)$ (2.4.9) satisfy the differential
equation

$-xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),0}(x)+\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),0}(x)=x+\hat{a},$
(2.4.21)

and for $l\geqslant 2$, the general differential equation

$xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),l}(x)-\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),l}(x)\\\
=\left[x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+4x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+2x\frac{\mathrm{d}}{\mathrm{d}x}\right]W_{1}^{(L),l-2}(x).$
(2.4.22)

###### Proof.

Consider equation (2.3.44) with $\beta=2$. In it, set $a=\hat{a}N$ and map
$x\mapsto\kappa Nx$ to obtain a differential equation for
$\tilde{W}_{1}^{(L)}(x)\big{|}_{a=\hat{a}N}$. Substitute in the expansion
(2.4.9) and equate terms of equal order in $N$ to extract differential
equations (2.4.21), (2.4.22). ∎

In the $\kappa=\beta/2=1$, $a=\hat{a}N$ setting considered here, the operator
$\mathcal{D}_{N,2}^{(L)}\big{|}_{x\mapsto\kappa Nx}$ (2.3.41) is even in $N$
while the right-hand side of differential equation (2.3.44) under the mapping
$x\mapsto\kappa Nx$ is odd in $N$. It follows that $\tilde{W}_{1}^{(L)}(x)$
(2.4.6) is an odd function of $N$, so $W_{1}^{(L),k}(x)=0$ when $k$ is odd.
Thus, differential-difference equation (2.4.22) holds vacuously for odd $l$,
and should otherwise be interpreted as a recursion over even $l$. Note also
that the vanishing of $W_{1}^{(L),k}(x)$ for $k$ odd implies that the spectral
moments $\tilde{m}_{k}^{(L)}$ (recall §1.1.1) of the scaled eigenvalue density
$\tilde{\rho}^{(L)}(\lambda;N,2)$ are hence even functions of $N$, in keeping
with the second point of Remark 2.1. In the $\beta=1,4$ analogue of
Proposition 2.13 given below, which is obtained by repeating the above proof
starting with the $\beta=1,4$ cases of equation (2.3.44) and taking
$a=\hat{a}\kappa N$, such phenomena are not seen since $W_{1}^{(L),k}(x)$ is
non-zero for all $k\in\mathbb{N}$.

######  Proposition 2.14.

Retain the definitions of $h$, $\hat{a}$ and $y_{(L)}$ from Propositions 2.1
and 2.13. For $\beta=1$ and $4$, the expansion coefficients of the scaled
resolvent $\tilde{W}_{1}^{(L)}(x;N,\beta)$ satisfy the differential equations

$-xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),0}(x)+\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),0}(x)=x+\hat{a},$
(2.4.23)

$xy_{(L)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),1}(x)-y_{(L)}^{2}\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),1}(x)\\\
=4h\hat{a}xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),0}(x)+2h\hat{a}\left[x^{2}-3(\hat{a}+2)x+2\hat{a}^{2}\right]W_{1}^{(L),0}(x)\\\
+2h\left[x(x-1)+2\hat{a}x+2\hat{a}^{2}\right],$ (2.4.24)
$xy_{(L)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),2}(x)-y_{(L)}^{2}\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),2}(x)\\\
=4h\hat{a}xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),1}(x)+2h\hat{a}\left[x^{2}-3(\hat{a}+2)x+2\hat{a}^{2}\right]W_{1}^{(L),1}(x)\\\
+\tfrac{5}{2}x^{3}y_{(L)}^{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(L),0}(x)+x^{2}\left[8x^{2}-19(\hat{a}+2)x+11\hat{a}^{2}\right]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(L),0}(x)\\\
+x\left[2x^{2}-6(\hat{a}+2)x+5\hat{a}^{2}\right]\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),0}(x)\\\
+2\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),0}(x)-2(\hat{a}+x),$
(2.4.25)

and for $l\geqslant 3$, the general differential equation

$xy_{(L)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),l}(x)-y_{(L)}^{2}\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),l}(x)\\\
=4h\hat{a}xy_{(L)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),l-1}(x)+2h\hat{a}\left[x^{2}-3(\hat{a}+2)x+2\hat{a}^{2}\right]W_{1}^{(L),l-1}(x)\\\
+\tfrac{5}{2}x^{3}y_{(L)}^{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(L),l-2}(x)+x^{2}\left[8x^{2}-19(\hat{a}+2)x+11\hat{a}^{2}\right]\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(L),l-2}(x)\\\
+x\left[2x^{2}-6(\hat{a}+2)x+5\hat{a}^{2}\right]\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(L),l-2}(x)+2\left[(\hat{a}+2)x-\hat{a}^{2}\right]W_{1}^{(L),l-2}(x)\\\
-h\hat{a}x\left[5x^{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+22x\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+14\frac{\mathrm{d}}{\mathrm{d}x}\right]W_{1}^{(L),l-3}(x)\\\
-\left[x^{5}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+10x^{4}\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}+22x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+4x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-4x\frac{\mathrm{d}}{\mathrm{d}x}\right]W_{1}^{(L),l-4}(x),$
(2.4.26)

where we set $W_{1}^{(L),-1}:=0$.

Before moving onto the Cauchy ensembles, we present the differential equations
characterising the expansion (2.4.10) in the JUE case with $a=\hat{a}\kappa
N$, $b=\hat{b}\kappa N$ as discussed immediately prior to Proposition 2.13. We
do not display the analogous results in the JOE and JSE cases due to their
(relatively speaking) unwieldy forms, but note that they can be derived from
equation (2.3.14) following the same proof as below.

######  Proposition 2.15.

Let
$y_{(J)}:=\sqrt{(\hat{a}+\hat{b}+2)^{2}x^{2}-2(\hat{a}+2)(\hat{a}+\hat{b})x+\hat{a}^{2}}$
with $\hat{a},\hat{b}$ constant in $N$. The expansion coefficients specified
by equation (2.4.10) in the JUE case with $a=\hat{a}\kappa N$ and
$b=\hat{b}\kappa N$ satisfy the differential equation

$x(x-1)y_{(J)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(J),0}(x)\\\
-\left[\hat{a}^{2}(1-x)^{3}-\hat{a}(\hat{b}+2)x(1-x)(1-2x)-(\hat{b}+2)^{2}x^{3}+2(\hat{b}+1)(3x^{2}+x)\right]W_{1}^{(J),0}(x)\\\
=(\hat{a}+\hat{b}+1)(\hat{a}(1-x)+\hat{b}x),$ (2.4.27)

and for $l\geqslant 2$, the general differential equation

$x(1-x)y_{(J)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(J),l}(x)\\\
+\left[\hat{a}^{2}(1-x)^{3}-\hat{a}(\hat{b}+2)x(1-x)(1-2x)-(\hat{b}+2)^{2}x^{3}+2(\hat{b}+1)(3x^{2}+x)\right]W_{1}^{(J),l}(x)\\\
=x^{3}(1-x)^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(J),l-2}(x)+4x^{2}(1-x)^{2}(1-2x)\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(J),l-2}(x)\\\
-2x(1-x)(7x(1-x)-1)\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(J),l-2}(x)-2x(1-x)(1-2x)W_{1}^{(J),l-2}(x).$
(2.4.28)

###### Proof.

Substitute expansion (2.4.10) into equation (2.3.8) and set $a=\hat{a}N$,
$b=\hat{b}N$. Equating terms of equal order in $N$ then produces the sought
differential equations. ∎

Similar to Proposition 2.13, in the $\beta=2$, $a=\hat{a}N$, $b=\hat{b}N$
setting, $W_{1}^{(J),k}(x)$ vanishes for odd $k$, and the differential
equations of Proposition 2.15 constitute a recursion over even $l$.

##### Differential equations for $W_{1}^{(Cy),l}(x)$ in the symmetric case
$\alpha$ real

The procedure outlined above extends to the Cauchy ensembles in the expected
manner. Having already set $\alpha=\hat{\alpha}\kappa N$ (recall that
$\eta=-\kappa(N-1)-1-\alpha$) to ensure that the global scaled eigenvalue
density $\tilde{\rho}^{(Cy)}(\lambda)$ has the necessary properties, we
simplify further by constraining $\hat{\alpha}$ to be a real and positive
constant. This decision to consider only the symmetric Cauchy ensembles is
simply due to the fact that the equivalent results in the non-symmetric case
are cumbersome to display; they can however be derived in the same way. Thus,
setting $\alpha=\hat{\alpha}\kappa N$ in equation (2.3.22) with
$\hat{\alpha}={\rm O}(1)$ in $N$, substituting in the expansion (2.4.11) for
$\tilde{W}_{1}^{(Cy)}(x)$, and then collecting terms of like order in $N$, we
obtain the following two propositions.

######  Proposition 2.16.

In the Cauchy weight, set $\eta=-\kappa(N-1)-1-\alpha$,
$\alpha=\hat{\alpha}\kappa N$ with $\hat{\alpha}$ positive real and constant
in $N$. Moreover, let
$y_{(Cy)}:=\sqrt{\hat{\alpha}^{2}x^{2}-2\hat{\alpha}-1}$. Then, the expansion
coefficients of the symmetric CyUE scaled resolvent
$\tilde{W}_{1}^{(Cy)}(x;N,2)\big{|}_{\hat{\alpha}\in\mathbb{R}_{+}}$ (2.4.11)
satisfy the differential equation

$-(1+x^{2})y_{(Cy)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),0}(x)-\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),0}(x)=(\hat{\alpha}+1)(2\hat{\alpha}+1),$
(2.4.29)

and for $l\geqslant 2$, the general differential equation

$4(1+x^{2})y_{(Cy)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),l}(x)+4\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),l}(x)\\\
=(1+x^{2})^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(Cy),l-2}(x)+8x(1+x^{2})^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(Cy),l-2}(x)\\\
+2(7x^{2}+3)(1+x^{2})\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),l-2}(x)+4x(1+x^{2})W_{1}^{(Cy),l-2}(x).$
(2.4.30)

As with the other classical unitary ensembles, $\tilde{W}_{1}^{(Cy)}(x)$ is an
odd function of $N$ when $\beta=2$, so $W_{1}^{(Cy),k}(x)=0$ when $k$ is odd.
It can thus be easily checked that differential-difference equation (2.4.30)
trivially holds true for odd values of $l$ and is otherwise to be interpreted
as a recursion over even $l$.

######  Proposition 2.17.

Retain the notation of Proposition 2.16. For $\beta=1$ and $4$, the expansion
coefficients of the scaled resolvent
$\tilde{W}_{1}^{(Cy)}(x;N,\beta)\big{|}_{\hat{\alpha}\in\mathbb{R}_{+}}$
satisfy the differential equations

$-(1+x^{2})y_{(Cy)}^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),0}(x)-\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),0}(x)=(\hat{\alpha}+1)(2\hat{\alpha}+1),$
(2.4.31)

$2(1+x^{2})y_{(Cy)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),1}(x)+2y_{(Cy)}^{2}\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),1}(x)\\\
=4(\kappa-1)\hat{\alpha}y_{(Cy)}^{2}(1+x^{2})^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),0}(x)+2(\kappa-1)\hat{\alpha}\left[2\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+3)^{2}+6\right]x(1+x^{2})W_{1}^{(Cy),0}(x)\\\
+(\kappa-1)\hat{\alpha}\left[(8\hat{\alpha}^{2}+9\hat{\alpha}+2)x^{2}+(2\hat{\alpha}+1)(5\hat{\alpha}+4)\right],$
(2.4.32)

$8(1+x^{2})y_{(Cy)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),2}(x)+8y_{(Cy)}^{2}\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),2}(x)\\\
=16(\kappa-1)\hat{\alpha}y_{(Cy)}^{2}(1+x^{2})^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),1}(x)\\\
+8(\kappa-1)\hat{\alpha}\left[2\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+3)^{2}+6\right]x(1+x^{2})W_{1}^{(Cy),1}(x)+10(\kappa-1)^{2}y_{(Cy)}^{2}(1+x^{2})^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(Cy),0}(x)\\\
+4(\kappa-1)^{2}\left[19\hat{\alpha}^{2}x^{2}-3\hat{\alpha}^{2}-22(2\hat{\alpha}+1)\right]x(1+x^{2})^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(Cy),0}(x)\\\
+4(\kappa-1)^{2}\left[29\hat{\alpha}^{2}x^{4}-2\hat{\alpha}^{2}x^{2}-44(2\hat{\alpha}+1)x^{2}+\hat{\alpha}^{2}-12(2\hat{\alpha}+1)\right](1+x^{2})\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),0}(x)\\\
+8(\kappa-1)^{2}\left[3\hat{\alpha}^{2}x^{4}-(\hat{\alpha}+8)^{2}x^{2}-4(2\hat{\alpha}+1)\right]W_{1}^{(Cy),0}(x)\\\
-4(\kappa-1)^{2}\left[(3\hat{\alpha}^{2}-1)x^{2}+9\hat{\alpha}^{2}+10\hat{\alpha}+3\right],$
(2.4.33)

and for $l\geqslant 3$, the general differential equation

$4(1+x^{2})y_{(Cy)}^{4}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),l}(x)+4y_{(Cy)}^{2}\left[\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+2)^{2}+2\right]xW_{1}^{(Cy),l}(x)\\\
=8(\kappa-1)\hat{\alpha}y_{(Cy)}^{2}(1+x^{2})^{2}\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),l-1}(x)\\\
+4(\kappa-1)\hat{\alpha}\left[2\hat{\alpha}^{2}x^{2}-(\hat{\alpha}+3)^{2}+6\right]x(1+x^{2})W_{1}^{(Cy),l-1}(x)\\\
+5(\kappa-1)^{2}y_{(Cy)}^{2}(1+x^{2})^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}W_{1}^{(Cy),l-2}(x)\\\
+2(\kappa-1)^{2}\left[19\hat{\alpha}^{2}x^{2}-3\hat{\alpha}^{2}-22(2\hat{\alpha}+1)\right]x(1+x^{2})^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}W_{1}^{(Cy),l-2}(x)\\\
+2(\kappa-1)^{2}\left[29\hat{\alpha}^{2}x^{4}-2\hat{\alpha}^{2}x^{2}-44(2\hat{\alpha}+1)x^{2}+\hat{\alpha}^{2}-12(2\hat{\alpha}+1)\right](1+x^{2})\frac{\mathrm{d}}{\mathrm{d}x}W_{1}^{(Cy),l-2}(x)\\\
+4(\kappa-1)^{2}\left[3\hat{\alpha}^{2}x^{4}-(\hat{\alpha}+8)^{2}x^{2}-4(2\hat{\alpha}+1)\right]W_{1}^{(Cy),l-2}(x)\\\
-(\kappa-1)^{3}\hat{\alpha}\left[5(1+x^{2})^{4}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+38x(1+x^{2})^{3}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\right.\\\
\left.+2(31x^{2}+15)(1+x^{2})^{2}\frac{\mathrm{d}}{\mathrm{d}x}+4x(4x^{4}+9x^{2}+5)\right]W_{1}^{(Cy),l-3}(x)\\\
-(\kappa-1)^{4}\bigg{[}(1+x^{2})^{5}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+20x(1+x^{2})^{4}\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}+(122x^{2}+29)(1+x^{2})^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}\\\
+4x(65x^{2}+46)(1+x^{2})^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+4(41x^{4}+56x^{2}+14)(1+x^{2})\frac{\mathrm{d}}{\mathrm{d}x}\\\
+8x(2x^{4}+4x^{2}+3)\bigg{]}W_{1}^{(Cy),l-4}(x)+4(\kappa-1)^{3}\hat{\alpha}x^{2}\chi_{l=3}+2(\kappa-1)^{4}(1-x^{2})\chi_{l=4},$
(2.4.34)

where we have set $W_{1}^{(Cy),-1}:=0$.

######  Remark 2.12.

  1. 1.

It can be observed that throughout this subsection, the differential equations
characterising the leading order terms $W_{1}^{0}(x)$ depend on the classical
weight $w(\lambda)$, but are otherwise the same for all $\beta$. Thus,
$W_{1}^{0}(x)$ is $\beta$-independent and use of the Sokhotski–Plemelj
inversion formula (1.1.25) tells us that the large $N$ limiting form
$\rho^{0}(\lambda)$ of $\tilde{\rho}(\lambda)$ also has this property. On the
other hand, with $a,b={\rm O}(N)$, $W_{1}^{1}(x)$ vanishes for $\beta=2$ but
can be seen to be non-zero for $\beta\neq 2$, in keeping with Note 1.1
following Proposition 1.2. In fact, solving the differential equations of
Propositions 2.11–2.15 with the boundary conditions (2.4.15), (2.4.16)
recovers expressions for $W_{1}^{0}(x)$ known from the general-$\beta$ works
[319], [142]. Applying the Sokhotski–Plemelj formula (1.1.25) then recovers
the expressions for $\rho^{0}(\lambda)$ given in Proposition 1.2 and its
$1/N,1/N^{2}$ corrections seen in [319], [142] (see also the earlier work
[17]).

  2. 2.

According to the second point of Remark 1.9 (more specifically equation
(1.2.88)), taking $a=\hat{a}\kappa N$ and $b=\hat{b}\kappa N$ with
$\hat{a},\hat{b}$ constant in $N$ is equivalent to setting
$\gamma_{1}=1+\hat{a}$ and $\gamma_{2}=1+\hat{b}$. Taking this into account,
we see that the auxiliary functions $y_{(w)}$ used in Propositions 2.11 to
2.17 are related to the large $N$ limiting forms given in Proposition 1.2. To
be precise, $y_{(G)}$, $y_{(L)}$, $y_{(J)}$, and $y_{(Cy)}$ are equal to the
Stieltjes transform of $\rho^{(G),0}(\lambda)$,
$2\lambda\rho^{(L),0}(\lambda)$, $2\lambda(1-\lambda)\rho^{(J),0}(\lambda)$,
and $(1+\lambda^{2})\rho^{(Cy),0}(\lambda)$, respectively. The significance of
their presence in the differential equations of this subsection is that for
$l>0$, the singularities of $W_{1}^{l}(x)$ lie exactly at the zeroes of
$y_{(w)}$, a fact well known from the viewpoint of topological recursion
[110].

We do not make explicit here the expressions for $W_{1}^{l}(x)$ nor
$\rho^{l}(x)$ in the Gaussian, Laguerre, or Jacobi cases, since these can be
found in the literature [17], [319], [142]. Instead, we demonstrate the
structures discussed in the above remark in the Cauchy case.

######  Proposition 2.18.

Retain the notation of Proposition 2.16, with $\hat{\alpha}$ positive real and
constant in $N$. For $\beta=1,2$, and $4$, we have

$\displaystyle W_{1}^{(Cy),0}(x)$
$\displaystyle=\frac{(\hat{\alpha}+1)x-y_{(Cy)}}{1+x^{2}},$ (2.4.35)
$\displaystyle W_{1}^{(Cy),1}(x)$
$\displaystyle=\frac{(\kappa-1)\hat{\alpha}}{2}\left[\frac{1}{y_{(Cy)}}-\frac{\hat{\alpha}x}{y_{(Cy)}^{2}}\right],$
(2.4.36) $\displaystyle W_{1}^{(Cy),2}(x)$
$\displaystyle=\frac{(\kappa-1)^{2}\hat{\alpha}}{2}\left[\frac{\hat{\alpha}^{2}((4\hat{\alpha}^{2}+10\hat{\alpha}+5)x^{2}+2\hat{\alpha}+1)}{4y_{(Cy)}^{5}}-\frac{\hat{\alpha}(\hat{\alpha}^{2}+2\hat{\alpha}+1)x}{y_{(Cy)}^{4}}\right]$
$\displaystyle\qquad+\frac{\kappa}{8}\frac{\hat{\alpha}^{2}(2\hat{\alpha}+1)(1+x^{2})}{y_{(Cy)}^{5}},$
(2.4.37)

along with the large $N$ asymptotic for the smoothed eigenvalue density
(2.4.4)

$\tilde{\rho}^{(Cy)}(\lambda)\Big{|}_{\hat{\alpha}\in\mathbb{R}_{+}}=\rho^{(Cy),0}(\lambda)+\frac{\kappa-1}{\kappa
N}\left[\frac{\hat{\alpha}}{2\pi\sqrt{1+2\hat{\alpha}-\hat{\alpha}^{2}\lambda^{2}}}\chi_{|\lambda|<\sqrt{2\hat{\alpha}+1}/\hat{\alpha}}\right.\\\
\left.-\frac{1}{4}\delta(\lambda-\sqrt{2\hat{\alpha}+1}/\hat{\alpha})-\frac{1}{4}\delta(\lambda+\sqrt{2\hat{\alpha}+1}/\hat{\alpha})\right]+{\rm
O}\left(\frac{1}{N^{2}}\right),$ (2.4.38)

where $\delta(\lambda)$ is the Dirac delta and $\rho^{(Cy),0}(\lambda)$ is
specified by equation (1.2.19) with $\hat{\alpha}=\hat{\alpha}_{1}$ and
$\hat{\alpha}_{2}=0$.

###### Proof.

The general solution of equation (2.4.29), equivalently (2.4.31), is

$W_{1}^{(Cy),0}(x)=\frac{(\hat{\alpha}+1)x+Cy_{(Cy)}}{1+x^{2}},$

with the integration constant $C$ forced to equal $-1$ due to the boundary
condition (2.4.15). Similarly, the integration constants present in the
general solutions of equations (2.4.30), (2.4.32), and (2.4.33) must be zero
due to the boundary condition (2.4.16). Equation (2.4.38) is obtained by
applying the Sokhotski–Plemelj formula (1.1.19) to the series (2.4.11)
truncated to have two terms; note that the Stieltjes transform (1.1.18) of
$\delta(\lambda-\lambda_{0})$ is $1/(x-\lambda_{0})$. ∎

Before moving on to the soft and hard edge scaling regimes, let us remark that
the results contained in Proposition 2.18 are expected to hold for general
real $\beta>0$, in line with what is known about the other classical $\beta$
ensembles [319], [142].

####  2.4.2 Soft and hard edge scaled differential equations

In this subsection we study the eigenvalue densities of the classical matrix
ensembles after they have been shifted to be centred on either the largest or
smallest eigenvalue and scaled such that the mean spacing between this
eigenvalue and its neighbour is order unity. The large $N$ limits of the
eigenvalue densities recentred and scaled in this manner have one of two
universal forms, depending on whether the centring is performed at a soft or
hard edge; we denote these large $N$ limiting forms
$\rho^{(soft)}(\lambda;\beta)$ and $\rho^{(hard)}(\lambda;\beta)$,
respectively. Recalling from the discussion following Proposition 1.2,
centring on the largest (smallest) eigenvalue corresponds to a soft edge if
$\rho^{0}(\lambda)$ exhibits a square root profile at the upper (lower)
endpoint of its support, and a hard edge if it instead has an inverse square
root profile. Thus, some observations can be made from the contents of
Proposition 1.2 using equation (1.2.88): When the parameters $a,b,\alpha$ are
proportional to $N$, i.e., upon setting $a=\hat{a}\kappa N$, $b=\hat{b}\kappa
N$, and $\alpha=\hat{\alpha}\kappa N$ with $\hat{a},\hat{b},\hat{\alpha}$
constant in $N$, all of the edges of the classical $\beta$ ensembles are of
the soft type. If one instead takes $a={\rm O}(1)$, the lower edge of the
Laguerre and Jacobi $\beta$ ensembles is of the hard type. Likewise, if
$b={\rm O}(1)$, $\rho^{(J),0}(\lambda)$ has a hard edge at the upper endpoint
of its support. A point of interest we will observe is that while edge scaling
is a procedure local to the endpoint of support being centred on, it depends
on both parameters $a,b$ where applicable, even though $a$ ($b$) mainly
governs the behaviour of the lower (upper) edge.

The particular scalings required to ensure that the large $N$ limits of the
eigenvalue densities of the classical matrix ensembles share the same
universal forms can be a little complicated, so we (re)introduce some
notation. Recall from the definitions of $a,b,\gamma_{1},\gamma_{2}$ in
Propositions 1.1 and 1.2 and equation (1.2.88) contained in Remark 1.9 that

$\displaystyle\gamma_{1}$
$\displaystyle=\lim_{N\to\infty}\frac{M_{1}}{N}=1+\lim_{N\to\infty}\frac{a-\kappa+1}{\kappa
N},$ (2.4.39) $\displaystyle\gamma_{2}$
$\displaystyle=\lim_{N\to\infty}\frac{M_{2}}{N}=1+\lim_{N\to\infty}\frac{b-\kappa+1}{\kappa
N},$ (2.4.40)

where $M_{1},M_{2}$ are now interpreted as continuous parameters dependent on
$a,b$. Thus in the large $N$ limit, $\gamma_{1},\gamma_{2}=1$ when $a,b={\rm
O}(1)$, while if we set $a=\hat{a}\kappa N$ and $b=\hat{b}\kappa N$ with
$\hat{a},\hat{b}={\rm O}(1)$, we have $\gamma_{1}=1+\hat{a}$ and
$\gamma_{2}=1+\hat{b}$. Recall too from Proposition 1.2 that
$\lambda_{\pm}^{(MP)},\lambda_{\pm}^{(Wac)},\lambda_{\pm}^{(Cy)}$ denote the
endpoints of the supports of
$\tilde{\rho}^{(L)}(\lambda),\tilde{\rho}^{(J)}(\lambda),\tilde{\rho}^{(Cy)}(\lambda)$,
respectively; in the following, we only consider the symmetric Cauchy
ensembles, so we set $\hat{\alpha}\in\mathbb{R}$ and
$\lambda_{\pm}^{(Cy)}=\pm\sqrt{2\hat{\alpha}+1}/\hat{\alpha}$. Let us also
define

$\gamma_{3}:=\frac{(\hat{\alpha}+1)^{2}}{\hat{\alpha}^{3}}$ (2.4.41)

for convenience and

$\displaystyle q^{2}$ $\displaystyle:=\frac{M_{1}}{M_{1}+M_{2}},\qquad
r^{2}:=\frac{M_{2}}{M_{1}+M_{2}},$ (2.4.42) $\displaystyle\tilde{q}^{2}$
$\displaystyle:=\frac{N}{M_{1}+M_{2}},\qquad\tilde{r}^{2}:=1-\tilde{q}^{2},$
(2.4.43) $\displaystyle u_{N}$
$\displaystyle:=\left[\frac{qr\tilde{q}\tilde{r}\sqrt{M_{1}+M_{2}}}{\tilde{q}\tilde{r}(q^{2}-r^{2})+qr(\tilde{q}^{2}-\tilde{r}^{2})}\right]^{4/3}$
(2.4.44)

following [178].

##### Differential equations characterising the soft edge

We now present differential equations satisfied by $\rho^{(soft)}(\lambda)$
for $\beta\in\\{2/3,1,2,4,6\\}$.

######  Theorem 2.3.

Define the soft edge limiting forms of the differential operators
$\mathcal{D}_{N,\beta}$ introduced in Section 2.3 as

$\mathcal{D}_{\beta}^{(soft)}=\begin{cases}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}-4x\frac{\mathrm{d}}{\mathrm{d}x}+2,&\beta=2,\\\
\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}-10\kappa
x\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+6\kappa\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+16\kappa^{2}x^{2}\frac{\mathrm{d}}{\mathrm{d}x}-8\kappa^{2}x,&\beta=1,4,\\\
3\frac{\mathrm{d}^{7}}{\mathrm{d}x^{7}}-56\kappa
x\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+28\kappa\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}+\frac{784}{3}\kappa^{2}x^{2}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}\\\
\quad-208\kappa^{2}x\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-4\kappa^{2}(64\kappa
x^{3}-17)\frac{\mathrm{d}}{\mathrm{d}x}+128\kappa^{3}x^{2},&\beta=2/3,6,\end{cases}$
(2.4.45)

where we recall that $\kappa=\beta/2$. Then, for $\beta\in\\{2/3,1,2,4,6\\}$,
the Gaussian, Laguerre, Jacobi, and symmetric Cauchy $\beta$ ensembles’
eigenvalue densities satisfy the following differential equation in the soft
edge limit:

$\mathcal{D}_{\beta}^{(soft)}\,\rho^{(soft)}(x;\beta)=0.$ (2.4.46)

###### Proof.

Make the change of variables [115]
$x\mapsto\sqrt{\kappa}\left(\sqrt{2N}+\frac{x}{\sqrt{2}N^{1/6}}\right)$ in
equations (2.0.3) and (2.3.53). Then, multiply through by $N^{-1/2}$ for
$\beta=2$, $N^{-5/6}$ for $\beta=1$ and $4$, or $N^{-7/6}$ for $\beta=2/3$ and
$6$. Equating terms of order one then yields equation (2.4.46) above, while
all other terms vanish in the $N\rightarrow\infty$ limit. ∎

In the case $\beta=2$, the differential equation (2.4.46) satisfied by the
soft edge density has been isolated in the earlier works of Brack et al. [50,
Eq. (C.2) with $\hbar^{2}/m=1$, $\lambda_{M}-V(r)=-{1\over 2}r$, $D=1$] and of
Dean et al. [77, Eq. (207) with $d=1$]. For $\beta=1,2$, and $4$, the above
proof can be replicated by instead considering the differential equations
(2.3.7), (2.3.13), (2.3.21) and (2.3.43) for the eigenvalue densities of the
Jacobi, symmetric Cauchy, and Laguerre ensembles, due to universality.
Explicitly, differential equation (2.4.46) is satisfied by the leading order
term of the following scaled densities in the large $N$ limit:

  * •

In the regime of the largest eigenvalue [115], [28], [132], [133],

$\rho^{(G)}\left(\sqrt{\kappa}\left(\sqrt{2N}+\delta^{(G)}+\frac{x}{\sqrt{2}N^{1/6}}\right);N,\beta\right),$
(2.4.47)

where $\delta^{(G)}={\rm o}(N^{-1/6})$ is an arbitrary parameter (the regime
of the smallest eigenvalue can be treated by exploiting the symmetry
$x\leftrightarrow-x$);

  * •

In the regime of the largest eigenvalue [115], [196], [121], [132], [133],

$\rho^{(L)}\left(\kappa\left(\lambda_{+}^{(MP)}N+\delta_{+}^{(L,soft)}+\frac{(\sqrt{\gamma_{1}}+1)^{4/3}}{\sqrt{\gamma_{1}}}N^{1/3}x\right);N,\beta\right),$
(2.4.48)

where $\delta^{(L,soft)}_{\pm}={\rm o}(N^{1/3})$ is henceforth an arbitrary
parameter. Note that this choice of scaling is effective whether we set
$a=\hat{a}\kappa N$ or take $a$ to be constant in $N$. In the latter case, the
argument of (2.4.48) simplifies to
$\kappa(4N+\delta_{+}^{(L,soft)}+2^{4/3}N^{1/3}x)$;

  * •

In the regime of the smallest eigenvalue, having set $a=\hat{a}\kappa N$ with
$\hat{a}={\rm O}(1)$ [28], [121],

$\rho^{(L)}\left(\kappa\left(\lambda_{-}^{(MP)}N-\delta^{(L,soft)}_{-}-(\sqrt{\gamma_{1}}-1)^{4/3}\left(\frac{N}{\sqrt{\gamma_{1}}}\right)^{1/3}x\right);N,\beta\right);$
(2.4.49)

  * •

In the regime of the largest eigenvalue, with $b=\hat{b}\kappa N$ and
$\hat{b}={\rm O}(1)$ [197], [121], [178],

$\rho^{(J)}\left(\lambda_{+}^{(Wac)}+\delta^{(J,soft)}+\frac{qr\tilde{q}\tilde{r}}{u_{N}}x;N,\beta\right),$
(2.4.50)

where $\delta^{(J,soft)}={\rm o}(N^{-2/3})$ is arbitrary and
$q,r,\tilde{q},\tilde{r},u_{N}$ are defined in (2.4.42)–(2.4.44). Like with
(2.4.48), this scaling is effective regardless of whether $a$ is constant or
linear in $N$ (we must take $b={\rm O}(N)$ to ensure that this is actually a
soft edge). The regime of the smallest eigenvalue can be treated by using the
symmetry $(x,a,b)\leftrightarrow(1-x,b,a)$;

  * •

In the regime of the largest eigenvalue, having set $\alpha=\hat{\alpha}\kappa
N$ with $\hat{\alpha}$ positive real and constant in $N$,

$\rho^{(Cy)}\left(\lambda_{+}^{(Cy)}+\delta^{(Cy)}+\left(\frac{\gamma_{3}^{2}}{2\lambda_{+}^{(Cy)}N^{2}}\right)^{1/3}x;N,\beta\right),$
(2.4.51)

where $\delta^{(Cy)}={\rm o}(N^{-2/3})$ is an arbitrary parameter and
$\gamma_{3}=(\hat{\alpha}+1)^{2}/\hat{\alpha}^{3}$ (2.4.41).

######  Remark 2.13.

For even $\beta$, $\rho^{(soft)}(x)$ has an explicit representation as a
$\beta$-dimensional integral due to [81], while [140] provides alternate forms
for $\beta=1,2$, and $4$. To date, there is no explicit functional form for
$\rho^{(soft)}(x)$ when $\beta=2/3$. On the other hand, [90] shows for all
$\kappa=\beta/2>0$ that for the first two leading orders as
$x\rightarrow\infty$,

$\rho^{(soft)}(x)\propto\frac{\exp\left(-4\kappa
x^{3/2}/3\right)}{x^{3\kappa/2}},$

which is an extension of the even-$\beta$ result of [120],

$\rho^{(soft)}(x)\underset{x\rightarrow\infty}{\sim}\frac{1}{\pi}\frac{\Gamma(1+\kappa)}{(8\kappa)^{\kappa}}\frac{\exp\left(-4\kappa
x^{3/2}/3\right)}{x^{3\kappa/2}}.$ (2.4.52)

This result is consistent with the differential equations of Theorem 2.3. So
too is the result [81],

$\rho^{(soft)}(x)\underset{x\rightarrow-\infty}{\sim}\frac{\sqrt{|x|}}{\pi},\quad\beta\in
2\mathbb{N}.$ (2.4.53)

Likewise, Theorem 2.4 below is consistent with the result [116],

$\rho^{(hard)}(x)\underset{x\rightarrow\infty}{\sim}\frac{1}{2\pi\sqrt{x}},\quad\beta\in
2\mathbb{N}.$ (2.4.54)

Note that the asymptotic forms (2.4.52), (2.4.53), and (2.4.54) respectively
capture the facts that moving past the soft edge results in exponential decay,
moving from the soft edge into the bulk results in a square root profile, and
moving from the hard edge into the bulk shows a square root singularity.

##### Differential equations characterising the hard edge

We now give the hard edge analogue of Theorem 2.3 for $\beta\in\\{1,2,4\\}$.
For uniformity across the Laguerre and Jacobi ensembles, we study the lower
edge centred at $x=0$ (the hard edge at $x=1$ for the Jacobi ensemble can be
studied by interchanging $x$ with $1-x$ and $a$ with $b$). In contrast to the
soft edge scaling, the differential equations characterising the hard edge are
seen to depend on the $a$ paramater, which is taken to be constant in $N$.

######  Theorem 2.4.

Define the hard edge limiting forms of the differential operators
$\mathcal{D}_{N,\beta}$ specified in Section 2.3 as

$\mathcal{D}_{\beta}^{(hard)}=\begin{cases}x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+4x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\left[x-a^{2}+2\right]x\frac{\mathrm{d}}{\mathrm{d}x}+\tfrac{1}{2}x-a^{2},&\beta=2,\\\
4x^{5}\frac{\mathrm{d}^{5}}{\mathrm{d}x^{5}}+40x^{4}\frac{\mathrm{d}^{4}}{\mathrm{d}x^{4}}+\left[10\kappa
x-5\tilde{a}+88\right]x^{3}\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}&\\\
\quad+\left[38\kappa
x-22\tilde{a}+16\right]x^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}&\\\
\quad+\left[\left(2\kappa x-\tilde{a}\right)^{2}+12\kappa
x-14\tilde{a}-16\right]x\frac{\mathrm{d}}{\mathrm{d}x}&\\\ \quad+(2\kappa
x-\tilde{a})(\kappa x-\tilde{a})-4\kappa x,&\beta=1,4,\end{cases}$ (2.4.55)

where we retain the definition of $\tilde{a}$ given in Theorem 2.2. Then, for
$\beta\in\\{1,2,4\\}$, the hard edge scaled eigenvalue densities of the
Laguerre and Jacobi $\beta$ ensembles satisfy the differential equation

$\mathcal{D}_{\beta}^{(hard)}\,\rho^{(hard)}(x;\beta)=0.$ (2.4.56)

###### Proof.

Since the Gaussian ensembles do not exhibit a hard edge, we turn to the
Laguerre ensembles as they are the next simplest to work with. Thus, we begin
by changing variables [115], [256] $x\mapsto\kappa x/(4N)$ in equation
(2.3.43). Equating terms of order one yields the differential equation
(2.4.56) above, and we note that all other terms are ${\rm O}(\frac{1}{N})$. ∎

As in the soft edge case, there is a little bit of freedom in the choice of
scaling used in the above proof, and the proof itself can be formulated in
terms of the differential equations (2.3.7) and (2.3.13) characterising the
eigenvalue densities of the Jacobi ensembles. To be precise, the differential
equation (2.4.56) above is satisfied by the leading order term of the
following scaled densities in the large $N$ limit:

  * •

With $\delta^{(L,hard)}={\rm o}(N)$ an arbitrary parameter [115], [256],
[134],

$\rho^{(L)}\left(\frac{\kappa x}{4N+\delta^{(L,hard)}};\beta\right);$ (2.4.57)

  * •

With $\delta^{(J,hard)}={\rm o}(N)$ also an arbitrary parameter,

$\rho^{(J)}\left(\frac{x}{N(4\gamma_{2}N+\delta^{(J,hard)})};\beta\right).$
(2.4.58)

While $a$ must be constant in $N$, applying this scaling to the differential
equations of Section 2.3 recovers Theorem 2.4 regardless of whether we take
$b$ to be proportional to $N$ or constant.

##### Optimal scaling at the soft and hard edges

It has previously been observed in [132] that the differential equation
characterisation of the soft edge scaled eigenvalue density can be extended to
similarly characterise the optimal leading order correction term. The latter
is obtained by a tuning of the $\delta^{(\,\cdot\,)}$ parameters in equations
(2.4.47)–(2.4.51) so as to obtain the fastest possible decay in $N$ of the
leading order correction, and thus the fastest possible convergence to the
limit. On this latter point, and considering the Gaussian case for
definiteness, we know from [133] that for $\beta$ even (at least),

$\displaystyle\hat{\rho}_{N,\beta}(x)$
$\displaystyle:={\sqrt{\kappa}\over\sqrt{2}N^{7/6}}\rho^{(G)}\left(\sqrt{\kappa}\left(\sqrt{2N}+\delta^{(G)}+\frac{x}{\sqrt{2}N^{1/6}}\right);N,\beta\right)$
(2.4.59)
$\displaystyle\;=\rho^{(soft)}(x;\beta)+\Big{(}\sqrt{2}N^{1/6}\delta^{(G)}-(1-1/\kappa)/(2N^{1/3})\Big{)}{\mathrm{d}\over\mathrm{d}x}\rho^{(soft)}(x;\beta)$
$\displaystyle\qquad+{\rm O}(N^{-2/3}),$ (2.4.60)

where the ${\rm O}(N^{-2/3})$ terms do not depend as simply on
$\rho^{(soft)}(x),{\mathrm{d}\over\mathrm{d}x}\rho^{(soft)}(x)$ as those made
explicit above. Thus, choosing $\delta^{(G)}=(1-1/\kappa)/(2\sqrt{2N})$ gives
the fastest convergence to the limit,

$\hat{\rho}_{N,\beta}(x)=\rho^{(soft)}(x;\beta)+{1\over
N^{2/3}}\zeta_{\beta}(x)+{\rm o}(N^{-2/3})$

for some $\zeta_{\beta}(x)$ which, for the values of $\beta$ permitting a
differential equation characterisation of $\rho^{(G)}(x)$ and
$\rho^{(soft)}(x)$, can itself be characterised as the solution of a
differential equation. The simplest case is $\beta=2$, when

$\zeta_{2}^{\prime\prime\prime}(x)-4x\zeta_{2}^{\prime}(x)+2\zeta_{2}(x)=x^{2}{\mathrm{d}\over\mathrm{d}x}\rho^{(soft)}(x;2)-x\rho^{(soft)}(x;2),$
(2.4.61)

which is an inhomogeneous generalisation of equation (2.4.46) for $\beta=2$.
For the particular Laguerre soft edge scaling (2.4.48), again considering
$\beta=2$, the analogue of equation (2.4.61) is given in [132, Eq. (4.19)],
with $a=\hat{a}\kappa N$ and $\delta^{(L,soft)}_{+}=0$.

More generally, multiplying the functions (2.4.47)–(2.4.51) by appropriate
scalars and powers of $N$ results in a function whose large $N$ asymptotic
expansion has leading order term given by $\rho^{(soft)}(x)$ and next to
leading order correction some function generically ${\rm O}(N^{-1/3})$ (when
$\beta=2$ and not all relevant parameters are constant, the correction term is
actually of order $N^{-2/3}$). Likewise, multiplying the functions (2.4.57)
and (2.4.58) by the correct pre-factors produces a function whose large $N$
limit is given by $\rho^{(hard)}(x)$ with correction term of generic order
$1/N$. In a similar fashion to the computation (2.4.60) above, proper tuning
of the $\delta^{(\,\cdot\,)}$ parameters seen in (2.4.47)–(2.4.51), (2.4.57),
(2.4.58) ensures that the correction terms are ${\rm O}(N^{-2/3})$ in the soft
edge case, and ${\rm O}(1/N^{2})$ in the hard edge case, which corresponds to
the scaled eigenvalue densities experiencing the optimal rate of convergence
to the limiting forms $\rho^{(soft)}(x)$ and $\rho^{(hard)}(x)$, respectively.

For $\beta$ an even integer, [133] shows that the scaling (2.4.48) becomes
optimal when we set

$\delta^{(L,soft)}_{+}=\begin{cases}\frac{2a}{\kappa},&a={\rm O}(1),\\\
\left(1-\frac{1}{\kappa}\right)\frac{\gamma_{1}-1}{2\sqrt{\gamma_{1}}},&a={\rm
O}(N).\end{cases}$ (2.4.62)

Likewise, [134] shows for general real $\beta>0$ and $a\in\mathbb{N}$ that the
scaling (2.4.57) is optimal upon setting $\delta^{(L,hard)}=2a/\kappa$, which
coincides with the value of $\delta^{(L,soft)}_{+}$ given above for the
$a={\rm O}(1)$ setting. We are able to extend these results by utilising
Theorems 2.3 and 2.4 as follows: Letting $\hat{\rho}_{N,\beta}(x)$ denote the
analogue of (2.4.59) constructed from either of the functions
(2.4.47)–(2.4.51), (2.4.57), or (2.4.58), we have that
$\hat{\rho}_{N,\beta}(x)$ is, at leading order, equal to either
$\rho^{(soft)}(x;\beta)$ or $\rho^{(hard)}(x;\beta)$. Then,
$\hat{\rho}_{N,\beta}(x)$ satisfies an edge-scaled version of one of the
differential equations of Section 2.3, while $\rho^{(soft)}(x;\beta)$ and
$\rho^{(hard)}(x;\beta)$ satisfy the differential equations of Theorems 2.3
and 2.4. Hence, the correction term
$\hat{\rho}_{N,\beta}(x)-\rho^{(soft)}(x;\beta)$, respectively
$\hat{\rho}_{N,\beta}(x)-\rho^{(hard)}(x;\beta)$, can be seen to satisfy a
certain differential equation. Analysis of this latter differential equation
reveals the order in $N$ of the correction term and its dependence on the
$\delta^{(\,\cdot\,)}$ parameters. Thus, for $\beta=1,2$, and $4$, we are able
to identify the values of $\delta^{(\,\cdot\,)}$ which optimally tune the
scalings (2.4.47)–(2.4.49), (2.4.51), (2.4.57), and (2.4.58). We confirm that
the values $\delta^{(G)}=(1-1/\kappa)/(2\sqrt{2N})$, $\delta^{(L,soft)}_{+}$
(2.4.62), and $\delta^{(L,hard)}=2a/\kappa$ recounted above from the works
[132], [133], [134] remain optimal when $\beta=1$ and/or $a>-1$ is allowed to
be general real. Moreover, we find that the scalings (2.4.49), (2.4.51) and
(2.4.58) are optimal when taking

$\displaystyle\delta^{(L,soft)}_{-}$
$\displaystyle=\left(1-\frac{1}{\kappa}\right)\frac{\gamma_{1}-1}{2\sqrt{\gamma_{1}}},$
(2.4.63) $\displaystyle\delta^{(Cy)}$
$\displaystyle=\left(1-\frac{1}{\kappa}\right)\frac{\gamma_{3}}{2\lambda_{+}^{(Cy)}N},$
(2.4.64) $\displaystyle\delta^{(J,hard)}$
$\displaystyle=\begin{cases}\frac{4(a+b)}{\kappa}-4\left(1-\frac{1}{\kappa}\right),&b={\rm
O}(1),\\\
\frac{2a}{\kappa}(\gamma_{2}+1)-4\left(1-\frac{1}{\kappa}\right),&b={\rm
O}(N).\end{cases}$ (2.4.65)

Note that the value of $\delta^{(L,soft)}_{-}$ given above is equal to that of
$\delta^{(L,soft)}_{+}$ given in equation (2.4.62) in the $a={\rm O}(N)$
setting. Moreover, the value of $\delta^{(J,hard)}$ given above for the
$b={\rm O}(1)$ case agrees with computations seen in the recent study [126,
App. A]. Due to computational limitations, we have not been able to identify
the optimal value of $\delta^{(J,soft)}$ for $\beta=1,4$, but note that its
value in the $\beta=1$ case can be surmised from [197] (which presumably
extends to the $\beta=4$ case through the duality principles of Lemma 1.4),
while for $\beta=2$, it should reduce to zero in the same fashion as
$\delta^{(G)},\delta^{(L,soft)}_{-},\delta^{(Cy)}$.

##  Chapter 3 Characterisations of the Moments and Cumulants

It was mentioned in §1.1.1 that the eigenvalue densities $\rho(\lambda)$ of
the random matrix ensembles studied in this thesis are fully characterised by
their spectral moments (though, one must keep in mind the caveat regarding the
Cauchy ensembles)

$m_{k}=\int_{\mathbb{R}}\lambda^{k}\rho(\lambda)\,\mathrm{d}\lambda=\left<\sum_{i=1}^{N}\lambda_{i}^{k}\right>,\quad
k\in\mathbb{N}.$ (3.0.1)

In the above, the latter presentation of the spectral moments is to be read as
an average with respect to the eigenvalue j.p.d.f. of the matrix ensemble of
interest. We begin this chapter with an application of the differential
equations given in Section 2.3 to the derivation of linear recurrences
characterising the spectral moments of the classical matrix ensembles (and the
Gaussian $\beta$ ensembles with $\beta=2/3,6$). We recall from Definition 1.5
that the classical matrix ensembles are exactly those whose eigenvalue
j.p.d.f.s are of the form

$p^{(w)}(\lambda_{1},\ldots,\lambda_{N};\beta)=\frac{1}{\mathcal{N}_{N,\beta}^{(w)}}\prod_{i=1}^{N}w(\lambda_{i})\,|\Delta_{N}(\lambda)|^{\beta},$
(3.0.2)

where $\beta=1,2$, or $4$,

$\Delta_{N}(\lambda)=\prod_{1\leqslant i<j\leqslant
N}(\lambda_{j}-\lambda_{i})$

is the Vandermonde determinant (1.1.9), and $w(\lambda)$ is a suitable
classical weight. In particular, we are interested in the spectral moments of
the Gaussian, Laguerre, Jacobi, symmetric shifted Jacobi, and symmetric Cauchy
ensembles, which are specified respectively by the weights (1.2.9), (1.2.29)

$\displaystyle w^{(G)}(\lambda)$ $\displaystyle=e^{-\lambda^{2}},$ (3.0.3)
$\displaystyle w^{(L)}(\lambda)$
$\displaystyle=\lambda^{a}e^{-\lambda}\chi_{\lambda>0},$ (3.0.4)
$\displaystyle w^{(J)}(\lambda)$
$\displaystyle=\lambda^{a}(1-\lambda)^{b}\chi_{0<\lambda<1},$ (3.0.5)
$\displaystyle w^{(sJ)}(\lambda)\big{|}_{a=b}$
$\displaystyle=(1-\lambda^{2})^{a}\chi_{-1<\lambda<1},$ (3.0.6) $\displaystyle
w^{(Cy)}(\lambda)\big{|}_{\alpha\in\mathbb{R}}$
$\displaystyle=(1+\lambda^{2})^{-\kappa(N-1)-1-\alpha};$ (3.0.7)

we recall that $\kappa:=\beta/2$ and the indicator function $\chi_{A}$ is
defined to equal one when $A$ is true and zero otherwise. The parameters
$a,b,\alpha$ are taken to be real throughout this chapter and, for convergence
issues, are further constrained such that $a,b>-1$ and $\alpha>-1/2$.

The aforementioned linear recurrences on the spectral moments are given in
§3.1.1, where we also show that these recurrence relations hold true for the
negative-integer moments ($m_{-k}$ with $k\in\mathbb{N}$), which have gained
interest primarily due to their connection to problems of quantum transport
[56], [240], [241], [71] and more recently due to duality principles relating
the negative-integer moments $m_{-k}$ to their counterparts $m_{k-1}$ [72]; in
fact, we show moreover that our moment recurrences are satisfied by a further
generalisation (see equation (3.1.1) forthcoming) of the $m_{k}$ concerning
complex values of $k$, which have recently been studied in [72], [23]. Having
linear recurrence relations on the spectral moments of the classical matrix
ensembles at hand means that we are able to derive so-called $1$-point
recursions [61] characterising the moment expansion coefficients defined in
Lemma 1.3. To be precise, we proceed in §3.1.2 by substituting the expansions

$\displaystyle m_{2k}^{(G)}$
$\displaystyle=\sum_{l=0}^{k}M_{k,l}^{(G)}N^{1+k-l},$ (3.0.8) $\displaystyle
m_{k}^{(L)}$ $\displaystyle=\sum_{l=0}^{k}M_{k,l}^{(L)}N^{1+k-l},$ (3.0.9)
$\displaystyle m_{k}^{(J)}$
$\displaystyle=\sum_{l=0}^{\infty}M_{k,l}^{(J)}N^{1-l}$ (3.0.10)

into the moment recurrences of §3.1.1 and equating terms of like order in $N$
to obtain $1$-point recursions on the $M_{k,l}^{(w)}$. Following [61, Defn.
1], this means that we find for each ensemble some integers
$i_{\mathrm{max}},j_{\mathrm{max}}$ and a non-trivial set of polynomials
$\\{q^{(w)}_{ij}(k)\\}_{0\leqslant i\leqslant i_{\mathrm{max}},0\leqslant
j\leqslant j_{\mathrm{max}}}$ such that whenever the involved terms are well-
defined, we have the equality

$\sum_{i=0}^{i_{\mathrm{max}}}\sum_{j=0}^{j_{\mathrm{max}}}q^{(w)}_{ij}(k)M^{(w)}_{k-i,l-j}=0.$
(3.0.11)

We present recursions of this type for the moment expansion coefficients of
the Gaussian $\beta$ ensembles with $\beta=2/3,6$, the (classical) Laguerre
ensembles, and the (symmetric shifted) Jacobi unitary ensemble, choosing to
forgo consideration of the remaining classical matrix ensembles in the
interest of brevity. However, we confirm here that applying our method to the
GUE recovers the celebrated Harer–Zagier recursion [174]

$4(k+1)M_{k,l}^{(GUE)}=4(2k-1)M_{k-1,l}^{(GUE)}+(k-1)(2k-1)(2k-3)M_{k-2,l-2}^{(GUE)}$
(3.0.12)

subject to the initial conditions $M_{0,0}^{(GUE)}=2M_{1,0}^{(GUE)}=1$,
$M_{1,1}^{(GUE)}=0$, and $M_{k,l}^{(GUE)}=0$ for all $k,l<0$, whereas treating
instead the GOE and GSE recovers the corresponding recursions of Ledoux given
in [224].

Evidently, the problem of computing the spectral moments of the classical
matrix ensembles is not a new one. Aside from recurrences on the moments
$m_{k}$ and their expansion coefficients $M_{k,l}$ known from the works [174],
[169], [223], [224], [319], [71], [72], [23] (which the computations of
Section 3.1 recover and/or supplement), the literature contains a range of
different approaches for studying the spectral moments of interest. In Section
3.2, we review some of these alternative characterisations of the spectral
moments of the classical matrix ensembles; in short, we survey parts of the
literature that show how the study of these spectral moments relates to skew-
orthogonal polynomial theory, symmetric function theory, and hypergeometric
orthogonal polynomials from the Askey scheme.

After the review of Section 3.2, the remainder of this chapter focuses on
elucidating how the spectral moments $m_{k}$, mixed moments (1.1.27)

$m_{k_{1},\ldots,k_{n}}=\left<\prod_{i=1}^{n}\operatorname{Tr}\,X^{k_{i}}\right>_{P(X)},\quad
k_{1},\ldots,k_{n}\in\mathbb{N},$ (3.0.13)

and mixed cumulants $c_{k_{1},\ldots,k_{n}}$ of particular random matrix
ensembles relate to the enumeration of ribbon graphs satisfying certain
constraints. Here, the term ‘ribbon graph’ needs to be formally introduced and
it is helpful to reiterate the definition of ‘mixed cumulants’. The mixed
cumulants $c_{\kappa_{i}}$ are defined implicitly by the moment-cumulants
relation (1.1.28)

$m_{k_{1},\ldots,k_{n}}=\sum_{K\vdash\\{k_{1},\ldots,k_{n}\\}}\prod_{\kappa_{i}\in
K}c_{\kappa_{i}},$ (3.0.14)

where we recall that $K\vdash\\{k_{1},\ldots,k_{n}\\}$ is read as “$K$ is a
partition of $\\{k_{1},\ldots,k_{n}\\}$”, which means that
$K=\\{\kappa_{i}\\}_{i=1}^{m}$ for some $1\leqslant m\leqslant n$ such that
the disjoint union $\kappa_{1}\sqcup\cdots\sqcup\kappa_{m}$ is equal to
$\\{k_{1},\ldots,k_{n}\\}$. For example, $K\vdash\\{4,6\\}$ if and only if $K$
is equal to $\\{\,\\{4,6\\}\,\\}$ or $\\{\,\\{4\\},\\{6\\}\,\\}$, so

$m_{4,6}=c_{4,6}+c_{4}\,c_{6}.$

As for ribbon graphs, we give here a pictorial definition that is well suited
to our purposes (cf. [211], [250], [219] for alternative but near-equivalent
definitions).

######  Definition 3.1.

Letting $k\in\mathbb{N}$, a $k$-ribbon graph is a collection of $n$ polygons
($0<n\leqslant 2k$) with labelled vertices and $k$ rectangles, referred to as
ribbons, satisfying the following properties:

  1. 1.

The number of edges of the polygons must sum to $2k$.

  2. 2.

For each ribbon, designate a pair of opposite edges to be referred to as ends
of the ribbon. Then, each of the $2k$ polygon edges must be identified, in a
topological sense, with exactly one ribbon-end, so that each ribbon connects
two polygon edges together.

The Euler genus of an orientable (non-orientable) connected ribbon graph is
the smallest $\tilde{g}\in\mathbb{N}$ such that the ribbon graph can be drawn
on a sphere with $\tilde{g}/2$ handles ($\tilde{g}$ cross-caps) without
intersecting with itself. In the orientable case, it is double the usual genus
$g$.

######  Note 3.1.

Since there are two ways to identify a pair of lines, depending on the
relative orientation of the lines, we allow our ribbons to have at most one
Möbius half-twist. We refer to the class of ribbon graphs with untwisted
ribbons as ‘(globally) orientable’, while the term ‘locally orientable’ refers
to the scenario where ribbons are allowed to be Möbius half-twisted. In the
literature, ribbon graphs are usually restricted to be of the first type,
while the latter type are referred to as Möbius graphs [251], [201], [57].

Figure 3.1: A $3$-ribbon graph constructed from a monogon, bigon, and
triangle, with a grey ribbon connecting the single edge of the monogon to the
left edge of the bigon, a Möbius half-twisted pink ribbon connecting the right
edge of the bigon to an edge of the triangle, and a blue ribbon connecting the
remaining two edges of the triangle together.

The connection between ribbon graphs and random matrix theory was first
explored by Brézin et al. in their 1978 work [54] (see also [36]), where they
were strongly influenced by ’t Hooft’s 1974 work [179] on the large $N$ limit
of gauge theories. Closely related ideas were seen independently in [326],
[304], [174] over the next decade, with Harer and Zagier’s 1986 work [174]
being of particular interest to us. Technically speaking, Harer and Zagier did
not look directly at ribbon graphs, but rather showed and utilised the fact
that $2^{k}M_{k,l}^{(GUE)}$ equals the number of ways of identifying pairs of
edges of a $2k$-gon to form a compact orientable surface of genus $l/2$ — of
course, this is equal to the number of orientable genus $l/2$ ribbon graphs
that can be built from a single $2k$-gon since the ribbons can be interpreted
as identifying the polygon edges at their ends. In 1997, Goulden and Jackson
[165] (see also [166], [251], [201]) gave the GOE analogue of the above
statement, which is that $4^{k}M_{k,l}^{(GOE)}$ equals the number of locally
orientable ribbon graphs of Euler genus $l$ that can be built from a single
$2k$-gon. Other related works from this time period include [271], [211],
[190], [287].

In §3.3.1, we give a detailed review of how the spectral moments of the GUE
and GOE relate to the ribbon graphs described above. Then, in §3.3.2, we
review the analogous statements for the LUE [82] and LOE [57]; the upshot here
is that $M_{k,l}^{(LUE)},M_{k,l}^{(LOE)}$ count the same ribbon graphs as
$M_{k,l}^{(GUE)},M_{k,l}^{(GOE)}$, respectively, except that the vertices of
the underlying $2k$-gons are alternately coloured black and red, and the edge
identifications induced by the ribbons respect this bicolouring. Our
arguments, which flow from the Isserlis–Wick theorem (see Theorem 3.1 of
Section 3.3), have been shown to extend to the GSE and LSE [251], [57], but we
do not explore this fact in order to keep our discussion concise;
$M_{k,l}^{(GSE)},M_{k,l}^{(LSE)}$ can roughly be ascribed the same
combinatorial meaning as $M_{k,l}^{(GOE)},M_{k,l}^{(LOE)}$ due to the
$\beta\leftrightarrow 4/\beta$ duality of Lemma 1.4. Finally, in §3.3.3, we
extend the arguments of §3.3.1 and §3.3.2 to produce ribbon graph
interpretations for the spectral moments of the Hermitised and antisymmetrised
matrix product ensembles defined in Section 1.3.

One of our motivations for supplying the review contained in §3.3.2 is that it
allows us to interpret the $1$-point recursions of §3.1.2 pertaining to the
Laguerre ensembles as solving combinatorial problems similar to that
considered by Harer and Zagier — unfortunately, we are currently unable to
find analogous applications for the other recursions given in §3.1.2 due to
complications that we outline in Section 3.3. As mentioned earlier, the
discussion of Section 3.3 does not focus solely on the spectral moments of the
random matrix ensembles at hand, but also their mixed moments and cumulants.
Indeed, we show therein how straightforward generalisations of the ribbon
graphs described above, now constructed from multiple polygons, relate to the
mixed moments and cumulants of interest. We show moreover that the relevant
mixed moments and cumulants are polynomial in $N$. A consequence of this is
that, upon appropriate scaling, we are able to assume that the connected
$n$-point correlators $W_{n}$ (1.1.29) of the Hermitised and antisymmetrised
matrix product ensembles have large $N$ (actually $N_{0}$) expansions of the
form (1.1.21) without needing to check the hypotheses of Theorem 1.2. The
existence of these large $N$ expansions is significant because they are
crucial to the loop equation analysis presented in Chapter 4.

###  3.1 Recurrence Relations for the Moments of the Classical Matrix
Ensembles

We now derive the aforementioned linear recurrence relations on the spectral
moments $m_{k}$ (3.0.1), and explain why they hold true when the $m_{k}$ are
instead taken to be the complex-$k$ generalised moments (following Cunden et
al. [72])

$m_{k}:=\int_{\mathbb{R}}|\lambda|^{k}\rho(\lambda)\,\mathrm{d}\lambda,\quad
k\in\\{k^{\prime}\in\mathbb{C}\,|\,m_{k^{\prime}}<\infty\\}.$ (3.1.1)

Note that these generalised moments agree with the spectral moments defined by
equation (3.0.1) for $k\in\mathbb{N}$ in the Jacobi and Laguerre cases, and
for $k\in 2\mathbb{N}$ in the Gaussian, symmetric shifted Jacobi, and
symmetric Cauchy cases — in the latter set of cases, $m_{k}=0$ for $k$ odd
according to equation (3.0.1), but not equation (3.1.1). Since the eigenvalue
densities studied in this section are either symmetric about the origin or
have support contained within the positive real line, the generalised moments
(3.1.1) can be interpreted as the spectral moments (we reserve the adjective
‘spectral’ for the moments defined by equation (3.0.1)) of the constrained
density $\rho(\lambda)\chi_{\lambda>0}$, with a possible factor of two.

The study of the former type of spectral moments is motivated by the fact that
they fully characterise the eigenvalue densities of the classical matrix
ensembles, as discussed in §1.1.1. Generalisation to negative integers $k$ was
first considered by Brouwer et al. [56] in the Laguerre case, where they
showed that the reciprocated eigenvalues of the Wigner–Smith time delay matrix
$Q$ are distributed according to the Laguerre ensemble with $a=\kappa N$ so
that $\left<\operatorname{Tr}\,Q^{k}\right>=m_{-k}^{(L)}|_{a=\kappa N}$ [71].
The significance of the Wigner–Smith matrix is that it relates to the study of
ballistic chaotic scattering, whereby its eigenvalues, referred to as proper
delay times, describe the amount of time incident particles spend in the
ballistic cavity during a scattering event, among other observables [33],
[56], [70], [71], [229], [240], [241]. The more recent extension to the
complex-$k$ moments (3.1.1) considered by Cunden et al. [72] relates to the
study of so-called spectral zeta functions of random matrices $X$,
$\zeta_{X}(k):=\operatorname{Tr}\,(X^{\dagger}X)^{-k}$ (note that if we take
$X$ to be the infinite-dimensional deterministic matrix
$\mathrm{diag}(1,\sqrt{2},\sqrt{3},\ldots)$, $\zeta_{X}(k)$ is then the
Riemann zeta function). It is shown in [72] that spectral zeta functions
satisfy a reflection symmetry across the line $\textrm{Re}(k)=1/2$ (much like
the Riemann zeta function), which implies reciprocity laws between the moments
$m_{-k}$ and $m_{k-1}$ of the LUE and JUE, for $k\in\mathbb{N}$; see Remark
3.1 within §3.2.3.

In §3.1.2, we return to the study of spectral moments (3.0.1) with
$k\in\mathbb{N}$ since then, it is known (recall Lemma 1.3) that
$m_{k}^{(G)},m_{k}^{(L)}$ are polynomial in $N$ and $m_{k}^{(J)}$ is a
rational function in $N$. Thus, the expansions (3.0.8)–(3.0.10) are valid and
we are able to derive $1$-point recursions (3.0.11) characterising the moment
expansion coefficients $M_{k,l}$. As mentioned earlier, $1$-point recursions
satisfied by these moment expansion coefficients have been studied in special
cases, with particular attention paid to their topological or combinatorial
interpretations [174], [224], [86]. These aspects are briefly discussed in
§3.1.2, with the highlighted point being that even though the moment expansion
coefficients $M_{k,l}$ may have combinatorial interpretations in certain cases
(fully detailed in Section 3.3), there are presently no such interpretations
for the $1$-point recursions satisfied by them.

####  3.1.1 Recurrences for the spectral moments

One may obtain recurrences for the spectral moments $m_{k}$ of the classical
matrix ensembles by recalling from §1.1.1 that the resolvent $W_{1}(x)$ acts
as a generating function for these moments, presuming that the moments are
well-defined for all $k\in\mathbb{N}$ or otherwise interpreting the large $x$
expansion of $W_{1}(x)$ in a formal sense:

$W_{1}(x)=\sum_{k=0}^{\infty}\frac{m_{k}}{x^{k+1}}.$ (3.1.2)

Substituting this series into the differential equations for $W_{1}(x)$ given
in Section 2.3 and then equating terms of equal order in $x$ gives relations
between the spectral moments. The equations obtained from terms of negative
order in $x$ give the upcoming recurrences on the spectral moments, while the
terms of order one and positive order in $x$ give the first few moments
required to run the recursions (the latter are also available in earlier
literature; see, e.g., [142] and references therein).

The recurrence relations derived from the above procedure enable the iterative
computation of the spectral moment $m_{k}$ for any positive integer $k$
starting with knowledge of $m_{0},\ldots,m_{p}$ for some small integer $p$.
However, these recurrences are actually valid for the complex-$k$
generalisations defined by equation (3.1.1), so long as all involved moments
$m_{k}$ are such that in the Gaussian and symmetric shifted Jacobi cases,
$\mathrm{Re}(k)>-1$; in the Jacobi and Laguerre cases, $\mathrm{Re}(k)>-a-1$;
and in the symmetric Cauchy case, $-1<\mathrm{Re}(k)<2\alpha+1$ (this is a
refinement of the constraint $-1<k<2\alpha_{1}+1$ discussed in §1.2.1). To
ascertain the validity of our moment recurrences in these more general
settings, we must provide an alternative proof to that outlined in the
previous paragraph. Thus, rather than the differential equations for
$W_{1}(x)$, we begin at the differential equations for the eigenvalue density
$\rho(x)$ given in Section 2.3. Multiplying both sides of these differential
equations by $|x|^{k}$ and then integrating over the support of the eigenvalue
density produces a relation between terms of the form
$\int_{\mathrm{supp}\,\rho}x^{p}|x|^{k}{\mathrm{d}^{n}\over\mathrm{d}x^{n}}\rho(x)\,\mathrm{d}x$
for positive integers $p,n$. The goal then is to use integration by parts to
reduce these terms to the form (3.1.1). This is done in a similar fashion to
what is shown in Appendix B, with the notable exception being that the domain
of integration must be split at the origin due to the absolute value sign in
the factor $|x|^{k}$ — in any case, all boundary terms arising from
integration by parts vanish, owing to our restrictions on $k$.

######  Proposition 3.1.

Recall the definitions of $\tilde{a},\tilde{b}$, and $\tilde{c}$ given in
Theorem 2.2. For $\beta=2$ and $k\in\mathbb{C}$ such that
$\mathrm{Re}(k)>2-a$, the moments (3.1.1) of the eigenvalue density of the
Jacobi ensemble satisfy the third order linear recurrence

$\sum_{l=0}^{3}d_{2,l}^{(J)}m_{k-l}^{(J)}=0,$ (3.1.3)

where

$\displaystyle d_{2,0}^{(J)}$
$\displaystyle=k\left[(a+b+2N)^{2}-(k-1)^{2}\right],$ $\displaystyle
d_{2,1}^{(J)}$
$\displaystyle=3k^{3}-11k^{2}-k\left[2(a+b+2N)^{2}+a^{2}-b^{2}-14\right]$
$\displaystyle\quad+3(a+b)(a+2N)+6(N^{2}-1),$ $\displaystyle d_{2,2}^{(J)}$
$\displaystyle=(2k-3)\left[2N(a+b+N)+ab\right]-(k-2)\left[3k^{2}-10k-3a^{2}+9\right],$
$\displaystyle d_{2,3}^{(J)}$
$\displaystyle=(k-3)\left[(k-2)^{2}-a^{2}\right].$

For $\beta=1,4$ and $k\in\mathbb{C}$ such that $\mathrm{Re}(k)>4-a$, the
moments of the Jacobi ensemble’s eigenvalue density satisfy the fifth order
linear recurrence

$\sum_{l=0}^{5}d_{4,l}^{(J)}m_{k-l}^{(J)}=0,$ (3.1.4)

where

$\displaystyle d_{4,0}^{(J)}$
$\displaystyle=k(\tilde{c}^{2}-(k-2)^{2})(\tilde{c}^{2}-(2k-1)^{2}),$
$\displaystyle d_{4,1}^{(J)}$
$\displaystyle=\frac{1}{2}(\tilde{c}^{2}-9)^{2}(5-6k)+\frac{1}{2}(\tilde{a}-\tilde{b})\left[(\tilde{c}^{2}-9)(5-4k)+2k(5(k-1)(k-5)+4k)\right]$
$\displaystyle\quad+(\tilde{c}^{2}-9)k\left[5(4k-3)(k-3)+2k\right]-4k^{2}(k-5)\left[5(k-2)(k-1)-2\right],$
$\displaystyle d_{4,2}^{(J)}$
$\displaystyle=\tilde{c}^{4}(3k-5)+\tilde{c}^{2}\left[\tfrac{1}{2}(\tilde{a}+\tilde{b})(2k-5)+5(\tilde{a}-\tilde{b})(k-2)\right]$
$\displaystyle\quad-\tilde{c}^{2}\left[30k^{3}-171k^{2}+339k-230\right]-\frac{1}{2}(\tilde{a}+\tilde{b})\left[5k^{3}-44k^{2}+129k-125\right]$
$\displaystyle\quad-\frac{1}{2}(\tilde{a}-\tilde{b})\left[35k^{3}-246k^{2}+581k-460\right]+\frac{1}{2}(2k-5)(\tilde{a}-\tilde{b})^{2}$
$\displaystyle\quad+40k^{4}(k-11)+1966k^{3}-4443k^{2}+5056k-2305,$
$\displaystyle d_{4,3}^{(J)}$
$\displaystyle=\frac{1}{2}\tilde{c}^{4}(5-2k)+\tilde{c}^{2}\left[\tfrac{1}{4}(\tilde{a}+\tilde{b})(25-8k)+\tfrac{1}{4}(\tilde{a}-\tilde{b})(45-16k)\right]$
$\displaystyle\quad+\tilde{c}^{2}\left[20k^{3}-155k^{2}+401k-345\right]+\frac{5}{4}(\tilde{a}+\tilde{b})\left[6k^{3}-62k^{2}+216k-253\right]$
$\displaystyle\quad+\frac{1}{4}(\tilde{a}-\tilde{b})\left[90k^{3}-806k^{2}+2436k-2485\right]+\frac{1}{4}(\tilde{a}^{2}-\tilde{b}^{2})(15-4k)$
$\displaystyle\quad+\frac{1}{4}(\tilde{a}-\tilde{b})^{2}(25-8k)-4k^{3}(10k^{2}-140k+789)+8923k^{2}-12600k+\tfrac{14125}{2},$
$\displaystyle d_{4,4}^{(J)}$
$\displaystyle=(k-4)\left[k^{3}+k^{2}-18k-\left(\tilde{c}^{2}-\tilde{b}-4k^{2}+29k-51\right)\left(5k^{2}-29k+40\right)\right]$
$\displaystyle\quad+\frac{1}{2}\tilde{a}^{2}(6k-25)+\frac{1}{2}\tilde{a}\left[(4k-15)\left(\tilde{c}^{2}-\tilde{b}-10k^{2}+76k-147\right)-2(k-5)\right],$
$\displaystyle d_{4,5}^{(J)}$
$\displaystyle=(k-5)\left[4(k-5)(k-4)-\tilde{a}\right]\left[\tilde{a}-(k-4)(k-2)\right].$

The sequences of spectral moments
$\\{m_{k}^{(J)}\\}_{k\in\mathbb{Z},\,k>-a-1}$ are fully determined for
$\beta=1,2,4$ by the above recurrence relations and the initial terms
$m_{0}^{(J)},m_{1}^{(J)},\ldots,m_{4}^{(J)}$, which can be computed through
MOPS [96] or via the methods presented in [242], [142] — according to the
discussion at the beginning of this subsection, the required initial terms are
also encoded within the right-hand sides of differential equations (2.3.8) and
(2.3.14).

The recurrence (3.1.3) for the moments $m_{k}^{(J)}$ of the JUE eigenvalue
density was recently given in [72, Prop. 4.8], wherein it is formulated as a
recurrence on the differences of moments $\Delta
m_{k}^{(J)}:=m_{k}^{(J)}-m_{k+1}^{(J)}$. The fact that this recurrence
relation can be naturally formulated in terms of the differences $\Delta
m_{k}^{(J)}$ is equivalent to the observation that
$\sum_{l=0}^{3}d_{2,l}^{(J)}=0$. Since $\sum_{l=0}^{5}d_{4,l}^{(J)}=0$, we see
that it is also reasonable to rewrite the recurrence relation (3.1.4) in terms
of the $\Delta m_{k}^{(J)}$. It turns out that the analogous recurrences for
the symmetric shifted Jacobi and symmetric Cauchy ensembles are most compactly
presented when written in terms of the differences and sums, respectively, of
even moments

$\displaystyle\mu_{k}^{(sJ)}$ $\displaystyle:=m_{2k+2}^{(sJ)}-m_{2k}^{(sJ)},$
(3.1.5) $\displaystyle\mu_{k}^{(Cy)}$
$\displaystyle:=m_{2k+2}^{(Cy)}+m_{2k}^{(Cy)}.$ (3.1.6)

Here, the moments $m_{k}$ are again defined by equation (3.1.1) with
$\rho(\lambda)$ being the eigenvalue density of the classical matrix ensemble
specified by either the constrained weight (3.0.6) or (3.0.7), respectively.
The fact that the $\mu_{k}^{(sJ)},\mu_{k}^{(Cy)}$ recurrences are simpler than
the corresponding recurrences on the $m_{2k}^{(sJ)},m_{2k}^{(Cy)}$ is in
keeping with the fact that the former are, respectively, the moments of
$r^{(sJ)}(x):=(1-x^{2})\rho^{(sJ)}(x)|_{a=b}$ and
$r^{(Cy)}(x):=(1+x^{2})\rho^{(Cy)}(x)|_{\alpha\in\mathbb{R}}$, which are
characterised by differential equations whose coefficients are of degree two
less than the corresponding coefficients of the differential equations
(2.3.17), (2.3.21) for $\rho^{(sJ)}(x)$ and $\rho^{(Cy)}(x)$; the differential
equations satisfied by $r^{(sJ)}(x)$ can be surmised from those satisfied by
$r^{(Cy)}(x)$, which are themselves given in §2.3.2 as equations (2.3.27),
(2.3.28).

######  Proposition 3.2.

Define $\mu_{k}^{(sJ)}$ by equation (3.1.5) and retain the definitions of
$a_{\beta},\tilde{a}$, and $\tilde{c}$ given in Theorem 2.2. For $\beta=2$ and
$k\in\mathbb{C}$ such that $\mathrm{Re}(k)>1/2$, we have the second order
linear recurrence

$(2k+4)\left[(2k+3)^{2}-4(a+N)^{2}\right]\mu_{k+1}^{(sJ)}-2(2k+1)\left[(2k+2)^{2}-2N(N+2a)\right]\mu_{k}^{(sJ)}\\\
+(2k+1)(2k)(2k-1)\mu_{k-1}^{(sJ)}=0.$ (3.1.7)

The initial condition determining the sequence
$\\{\mu_{k}^{(sJ)}|_{\beta=2}\\}_{k\in\mathbb{N}}$ is

$\mu_{0}^{(sJ)}={2N(a+N)(2a+N)\over 1-4(a+N)^{2}}.$ (3.1.8)

For $\beta=1,4$ and $k\in\mathbb{C}$ such that $\mathrm{Re}(k)>3/2$, we have
the fourth order linear recurrence

$\sum_{l=-2}^{2}d_{4,l}^{(sJ)}\mu_{k-l}^{(sJ)}=0,$ (3.1.9)

where

$\displaystyle d^{(sJ)}_{4,-2}$
$\displaystyle:=-4(2k+1)(2k)(2k-1)(2k-2)(2k-3),$ $\displaystyle
d^{(sJ)}_{4,-1}$
$\displaystyle:=(2k+1)(2k)(2k-1)\left[20\tilde{a}-5\tilde{c}^{2}+16k(4k+5)+77\right],$
$\displaystyle d^{(sJ)}_{4,0}$
$\displaystyle:=(2k+1)\Big{[}8\tilde{a}(5k+8)(2k+3)-\left(\tilde{c}^{2}-4\tilde{a}-30k^{2}-67k-42\right)^{2}$
$\displaystyle\qquad+516k^{4}+2292k^{3}+3653k^{2}+2534k+673\Big{]},$
$\displaystyle d^{(sJ)}_{4,1}$
$\displaystyle:=\tilde{c}^{2}\left[(\tilde{c}^{2}-4\tilde{a})(4k+7)-120k^{3}-656k^{2}-1210k-746\right]$
$\displaystyle\quad+\tilde{a}\left[160k^{3}+736k^{2}+1128k+580\right]$
$\displaystyle\quad+512k^{5}+4480k^{4}+16056k^{3}+29360k^{2}+27270k+10243,$
$\displaystyle d^{(sJ)}_{4,2}$
$\displaystyle:=-2(k+3)(\tilde{c}+4k+11)(\tilde{c}+2k+4)(\tilde{c}-2k-4)(\tilde{c}-4k-11).$

The initial conditions determining the sequences
$\\{\mu_{k}^{(sJ)}|_{\beta=1,4}\\}_{k\in\mathbb{N}}$ are

$\displaystyle\mu_{0}^{(sJ)}$
$\displaystyle=\frac{(\tilde{c}-1)\left(\tilde{c}+2a_{\beta}-3\right)\left(\tilde{c}-2a_{\beta}+1\right)}{8\tilde{c}(\tilde{c}-3)(1-\kappa)},$
(3.1.10) $\displaystyle\mu_{1}^{(sJ)}$
$\displaystyle=\frac{(\tilde{c}^{2}-5)(\tilde{c}-7)-4\tilde{a}(\tilde{c}-1)}{4(\tilde{c}^{2}-4)(\tilde{c}-7)}\mu_{0}^{(sJ)}.$
(3.1.11)

We note that the generalisation of recurrence (3.1.7) concerning the non-
symmetric shifted JUE corresponding to the weight
$w^{(sJ)}(\lambda)=(1-\lambda)^{a}(1+\lambda)^{b}\chi_{-1<\lambda<1}$ and
$\beta=2$ was first given by Ledoux in [223], albeit consideration was given
only to the $\mu^{(sJ)}_{k}$ with $k\in\mathbb{N}$. Moving on, let us observe
that the contents of Section 2.2, particularly Proposition 2.2, imply the
relation

$\mu_{k}^{(sJ)}\Big{|}_{a\mapsto-\kappa(N-1)-1-\alpha}=(-1)^{k-1}\mu_{k}^{(Cy)},$
(3.1.12)

where we recall that $\kappa=\beta/2$. A simple consequence of this relation
is that Proposition 3.2 leads to the following recurrence relations for the
sums of moments $\mu_{k}^{(Cy)}$.

######  Corollary 3.1.

Define $\mu_{k}^{(Cy)}$ by equation (3.1.6) and retain the definitions of
$d^{(sJ)}_{4,-2},d^{(sJ)}_{4,-1},\ldots,d^{(sJ)}_{4,2}$ listed in Proposition
3.2. For $\beta=2$ and $k\in\mathbb{C}$ such that
$1/2<\mathrm{Re}(k)<\alpha-3/2$, we have the second order linear recurrence
(recently given in [23])

$(2k+4)\left[(2k+3)^{2}-4\alpha^{2}\right]\mu_{k+1}^{(Cy)}+2(2k+1)\left[(2k+2)^{2}+2N(N+2\alpha)\right]\mu_{k}^{(Cy)}\\\
+(2k+1)(2k)(2k-1)\mu_{k-1}^{(Cy)}=0.$ (3.1.13)

The initial condition determining the sequence
$\\{\mu_{k}^{(Cy)}|_{\beta=2}\\}_{k\in\mathbb{N}}$ is

$\mu_{0}^{(Cy)}={2N\alpha(N+2\alpha)\over(2\alpha-1)(2\alpha+1)},\quad\alpha>1/2.$
(3.1.14)

For $\beta=1,4$ and $k\in\mathbb{C}$ such that
$3/2<\mathrm{Re}(k)<\alpha-5/2$, we have the fourth order linear recurrence

$\sum_{l=-2}^{2}d_{4,l}^{(Cy)}\mu_{k+l}^{(Cy)}=0,\qquad
d_{4,l}^{(Cy)}:=(-1)^{l-1}d_{4,l}^{(sJ)}\Big{|}_{a\mapsto-\kappa(N-1)-1-\alpha}.$
(3.1.15)

The initial conditions required to fully characterise the sequences
$\\{\mu_{k}^{(Cy)}|_{\beta=1,4}\\}_{k\in\mathbb{N}}$ are obtained by parsing
equations (3.1.10) and (3.1.11) through the relation (3.1.12).

In a similar vein to how the arguments of Section 2.2 enable us to simply
translate the recurrences of Proposition 3.2 for the $\mu_{k}^{(sJ)}$ into the
analogous recurrences on the $\mu_{k}^{(Cy)}$ given in Corollary 3.1, the
limiting procedure (1.2.23) described in Lemma 1.1 can be used to obtain
recurrences on the (generalised) spectral moments $m_{k}^{(L)}$ of the
Laguerre ensembles from the recurrences presented in Proposition 3.1. Indeed,
to derive the following recurrences on the $m_{k}^{(L)}$, one may circumvent
the strategy outlined at the beginning of this subsection by setting
$m_{k}^{(J)}=b^{-k}m_{k}^{(L)}$ in Proposition 3.1 and then taking the limit
$b\to\infty$ (cf. equation (2.3.40), in addition to Lemma 1.1).

######  Corollary 3.2.

Recall the definitions of $a_{\beta},N_{\beta}$, and $\tilde{a}$ given in
Theorem 2.2. For $\beta=2$ and $k\in\mathbb{C}$ such that
$\mathrm{Re}(k)>1-a$, the moments of the LUE eigenvalue density satisfy the
second order linear recurrence

$(k+1)m_{k}^{(L)}=(2k-1)(a+2N)m_{k-1}^{(L)}+(k-2)\left[(k-1)^{2}-a^{2}\right]m_{k-2}^{(L)}.$
(3.1.16)

For $\beta=1,4$ and $k\in\mathbb{C}$ such that $\mathrm{Re}(k)>3-a$, we have
the fourth order linear recurrence

$\sum_{l=0}^{4}d_{4,l}^{(L)}(\kappa-1)^{l}m_{k-l}^{(L)}=0,$ (3.1.17)

where

$\displaystyle d_{4,0}^{(L)}$ $\displaystyle=k+1,$ $\displaystyle
d_{4,1}^{(L)}$ $\displaystyle=(1-4k)(a_{\beta}+4N_{\beta}),$ $\displaystyle
d_{4,2}^{(L)}$
$\displaystyle=(1-k)(5k^{2}-11k+4)+(2k-3)\left[\tilde{a}+2(a_{\beta}+4N_{\beta})^{2}\right],$
$\displaystyle d_{4,3}^{(L)}$
$\displaystyle=(a_{\beta}+4N_{\beta})\left[(11-4k)\tilde{a}+10k^{3}-68k^{2}+146k-96\right],$
$\displaystyle d_{4,4}^{(L)}$
$\displaystyle=(k-4)\left[\tilde{a}-4(k-4)(k-3)\right]\left[\tilde{a}-(k-3)(k-1)\right].$

The initial terms $m_{0}^{(L)},m_{1}^{(L)},\ldots,m_{3}^{(L)}$ required to
fully determine the sequences $\\{m_{k}^{(L)}\\}_{k\in\mathbb{Z},\,k>-a-1}$ in
the orthogonal, unitary, and symplectic cases can be found in [242], [142].

It should come as no surprise that recurrence (3.1.16) on the moments of the
LUE eigenvalue density has already been given in the literature, seeing as
this is true of its JUE analogue (3.1.3), which is more involved. Indeed, it
was first obtained by Haagerup and Thorbjørnsen [169], where they considered
only the case $a,k\in\mathbb{N}$. A bit over a decade later, Cunden et al.
[71] showed that the LUE moment recurrence (3.1.16) holds for all integers
$k>-a-1$ with $a>-1$ a continuous real parameter. On the other hand,
recurrence (3.1.17) pertaining to the moments of the LOE and LSE can be seen
to be a homogeneous simplification of [71, Eq. (43)], wherein the
inhomogeneous terms depend on the moments of the LUE.

Another simple observation is that for each of $\beta=1,2$, and $4$, the
$m_{k}^{(L)}$ recurrences displayed in Corollary 3.2 involve one less term
than the corresponding $m_{k}^{(J)}$ recurrences given in Proposition 3.1 —
the terms in the recurrences are themselves drastically simpler, as well. This
is in keeping with the loss of parameter $b$. A similar simplification of
terms can be observed in the moment recurrences in the Gaussian case, again to
be understood as a consequence of the hierarchy implied by Lemma 1.1. However,
we do not see a reduction in the order of the moment recurrences when moving
from the Laguerre to Gaussian ensembles. Thus, the spectral (and generalised)
moments of the GUE satisfy a second order linear recurrence first derived by
Harer and Zagier in [174], while the spectral moments of the GOE and GSE are
characterised by fourth order linear recurrences given by Ledoux in [224]. We
do not present them here, but note (as was mentioned earlier) that the
strategies discussed in this section recover the recurrences of [174], [224],
with the added insight that they hold for complex $k$. Furthermore, our
methods allow us to derive linear recurrence relations for the $\beta=2/3$ and
$\beta=6$ Gaussian ensembles’ spectral moments, which we now give.

######  Proposition 3.3.

For $\beta=2/3,6$ and $k\in\mathbb{C}$ such that $\mathrm{Re}(k)>11$, the
moments of the Gaussian $\beta$ ensemble’s eigenvalue density satisfy the
sixth order linear recurrence

$\sum_{l=0}^{6}d_{6,l}^{(G)}\left(\frac{\kappa-1}{4}\right)^{l}m_{k-2l}^{(G)}=0,$
(3.1.18)

where

$\displaystyle d_{6,0}^{(G)}$ $\displaystyle=-4(k+2),$ $\displaystyle
d_{6,1}^{(G)}$ $\displaystyle=8(3k-1)(3N_{\beta}+2),$ $\displaystyle
d_{6,2}^{(G)}$
$\displaystyle=48(8-3k)N_{\beta}(3N_{\beta}+4)+49k^{3}-216k^{2}+92k+320,$
$\displaystyle d_{6,3}^{(G)}$
$\displaystyle=4(k-5)(3N_{\beta}+2)\left[24N_{\beta}(3N_{\beta}+4)-49k^{2}+304k-442\right],$
$\displaystyle d_{6,4}^{(G)}$
$\displaystyle=2(k-5)_{3}\left[294N_{\beta}(3N_{\beta}+4)-63k(k-6)-274\right],$
$\displaystyle d_{6,5}^{(G)}$ $\displaystyle=252(k-5)_{5}(3N_{\beta}+2),$
$\displaystyle d_{6,6}^{(G)}$ $\displaystyle=81(k-5)_{7},$

and we retain the definition $N_{\beta}=(\kappa-1)N$. Here,
$(x)_{n}=x(x-1)\cdots(x-n+1)$ is the falling Pochhammer symbol. The initial
terms $m_{0}^{(G)},m_{2}^{(G)},\ldots,m_{10}^{(G)}$ needed to determine the
sequences $\\{m_{k}^{(G)}|_{\beta=2/3,6}\\}_{k\in\mathbb{N}}$ are listed in
[319].

As we proceed to our discussion on $1$-point recursions, let us conclude this
subsection with a couple of general remarks. The recurrences given above
describe relations between the complex-$k$ generalised moments $m_{k}$ defined
by equation (3.1.1), so long as all moments involved are well-defined.
However, the recurrences hold true when the $m_{k}$ are taken to be
(integer-$k$) spectral moments as defined by equation (3.0.1), even though
these spectral moments fail to coincide with the generalised moments when
considering the Gaussian, symmetric Jacobi, or symmetric Cauchy ensembles,
with $k$ an odd integer. In these cases, the definitions (3.0.1) and (3.1.1)
disagree only in that, according to the first definition, $m_{k}=0$ when $k$
is odd — the relevant recurrences are then seen to hold trivially for odd
integers $k$ due to their homogeneous structure and the fact that they run
over every second integer-$k$ moment (e.g., recurrence (3.1.15) involves
$m_{k+6}^{(Cy)},m_{k+4}^{(Cy)},\ldots,m_{k-4}^{(Cy)}$).

The methods employed in [224] manifest a coupling between the GOE and GUE
moments, which does not arise naturally from our viewpoint. Likewise for the
coupling between the LOE and LUE moments implied by the recurrence [71, Eq.
(43)]. As made clear in [71], both couplings can be traced back to a
structural formula expressing the $\beta=1$ eigenvalue densities in terms of
their $\beta=2$ counterparts plus what can be regarded as rank one
corrections. This inter-relation was discussed briefly in §1.2.3, where we saw
that the context underlying it is that the skew-orthogonal polynomials
pertaining to the GOE and LOE can be constructed from the Hermite and Laguerre
orthogonal polynomials used to study the GUE and LUE, respectively. At
present, there is no evidence implying a relation connecting the moments of
the $\beta=6$ Gaussian ensemble to the moments of the GOE and/or GUE.

####  3.1.2 $1$-point recursions for the moment expansion coefficients

Following on from the preceding subsection, it is a simple matter to obtain
$1$-point recursions of the form (3.0.11) for the coefficients of the moment
expansions (3.0.8)–(3.0.10). In fact, all one needs to do is substitute the
moment expansions (3.0.8)–(3.0.10) into the recurrence relations of §3.1.1,
now considering $k$ large enough integers such that all involved moments are
well-defined according to definition (3.0.1), and then equate terms of equal
order in $N$. There is however another (instructive) avenue towards the sought
$1$-point recursions beginning at the differential-difference equations of
§2.4.1: In the large $N$ limit, the spectral moments
$m_{2k}^{(G)},m_{k}^{(L)}$, and $m^{(J)}_{k}$ do not converge, so the moment
expansions (3.0.8)–(3.0.10) may only be understood in a formal sense.
Rectifying this issue by introducing the scalings

$\displaystyle\tilde{m}_{2k}^{(G)}$
$\displaystyle:=\kappa^{-k}N^{-k-1}\,m_{2k}^{(G)},$ (3.1.19)
$\displaystyle\tilde{m}_{k}^{(L)}$
$\displaystyle:=\kappa^{-k}N^{-k-1}\,m_{k}^{(L)},$ (3.1.20)
$\displaystyle\tilde{m}_{k}^{(J)}$ $\displaystyle:=m_{k}^{(J)}/N,$ (3.1.21)

as is compliant with Definition 1.6 (that is,
$\tilde{m}_{k}=\int_{\mathbb{R}}\lambda^{k}\tilde{\rho}(\lambda)\,\mathrm{d}\lambda$),
we observe that the (scaled) resolvents $\tilde{W}_{1}^{(G)}(x)$ (2.4.5),
$\tilde{W}_{1}^{(L)}(x)$ (2.4.6), and $W_{1}^{(J)}(x)$ act as generating
functions for
$\\{\tilde{m}_{2k}^{(G)}\\}_{k\in\mathbb{N}},\\{\tilde{m}_{k}^{(L)}\\}_{k\in\mathbb{N}}$,
and $\\{\tilde{m}_{k}^{(J)}\\}_{k\in\mathbb{N}}$, respectively. Moreover, the
scaled moments have convergent large $N$ expansions (finite sums in the
Gaussian and Laguerre cases) of the form
$\tilde{m}_{k}=\sum_{l=0}^{\infty}\tilde{M}_{k,l}\,N^{-l}$ (1.1.32), with the
expansion coefficients $\tilde{M}_{k,l}$ of the scaled moments themselves
generated by the resolvent expansion coefficients $W_{1}^{l}(x)$ defined
implicitly in equations (2.4.8)–(2.4.10). Thus, substituting the large $x$
expansions

$\displaystyle W_{1}^{(G),l}(x)$
$\displaystyle=2^{l+1}\kappa^{l/2}\sum_{k=0}^{\infty}\tilde{M}^{(G)}_{k,l}\,x^{-k-1},$
$\displaystyle W_{1}^{(L),l}(x)$
$\displaystyle=\kappa^{l/2}\sum_{k=0}^{\infty}\tilde{M}^{(L)}_{k,l}\,x^{-k-1},$
$\displaystyle W_{1}^{(J),l}(x)$
$\displaystyle=\sum_{k=0}^{\infty}\tilde{M}^{(J)}_{k,l}\,x^{-k-1}$

into the differential-difference equations of §2.4.1 and then comparing terms
of like order in $x$ yields $1$-point recursions on the $\tilde{M}_{k,l}$
(recall the diagram displayed at the end of §1.1.1). To obtain equivalent
recursions for the unscaled moment expansion coefficients $M_{k,l}$, it is
enough to note from equations (3.1.19)–(3.1.21) that

$M_{k,l}^{(G)}=\kappa^{k}\,\tilde{M}_{2k,l}^{(G)},\qquad
M_{k,l}^{(L)}=\kappa^{k}\,\tilde{M}_{k,l}^{(L)},\qquad
M_{k,l}^{(J)}=\tilde{M}_{k,l}^{(J)}.$

Confirming agreement between the two derivations outlined above, flowing
separately from the results of §3.1.1 and §2.4.1, respectively, serves as a
consistency check. That aside, the upcoming $1$-point recursions are
themselves motivated by the fact that they enable efficient computation of the
expansion coefficients $M_{k,l}$, assuming knowledge of sufficiently many of
them for small $k,l\in\mathbb{N}$ — we do not provide the necessary initial
conditions here, but note that they can be computed through the strategy
discussed at the beginning of §3.1.1. Compared to the recurrences of §3.1.1,
which run over the spectral moments $m_{k}$, the $1$-point recursions are
faster at isolating the leading order behaviour of the $m_{k}$, in the large
$N$ limit. On the other hand, changing viewpoint to that of Section 3.3, the
$1$-point recursions are interesting because they relate enumerations of
ribbon graphs in a manner that has no obvious combinatorial interpretation; we
make only brief comments on the relevant ribbon graphs here, deferring full
discussion to Section 3.3.

As mentioned earlier, our $1$-point recursions agree with that of Harer and
Zagier (3.0.12) in the GUE case and of Ledoux [224] in the GOE and GSE cases.
Let us recall that the motivation for studying the former came from the
interpretation of $2^{k}M_{k,l}^{(GUE)}$ as the number of ways of gluing the
edges of a $2k$-gon to form a compact orientable surface of genus $l/2$ (cf.
Figure 3.5 of §3.3.1), while $4^{k}M_{k,l}^{(GOE)}$ is known [165], [166],
[251], [201] to be equal to the number of such gluings that form a compact
locally orientable surface of Euler characteristic $2-l$ (see Figure 3.9);
$M_{k,l}^{(GSE)}$ has been given equivalent interpretations in [251], [57],
which can be understood through the $\beta\leftrightarrow 4/\beta$ duality of
Lemma 1.4. We do not recount the GOE, GUE, and GSE $1$-point recursions here,
for brevity. Likewise, we do not report on the symmetric Cauchy ensembles, nor
the (symmetric shifted) Jacobi ensemble when $\beta\neq 2$; though these have
not been given in the literature before, they are relatively cumbersome to
display while being as equally easy to derive from the recurrences of §3.1.1
as the recursions given below. Our first result is hence on the Gaussian
$\beta$ ensembles with $\beta=2/3$ and $6$.

######  Proposition 3.4.

Expand the spectral moments of the Gaussian ensemble according to (3.0.8).
Then, for $\beta=2/3,6$ and $k\geqslant 6$,

$(k+1)M_{k,l}^{(G)}=\sum_{i=1}^{6}\sum_{j=0}^{i}\frac{(\kappa-1)^{2i-j}}{2^{i}}f_{i,j}M_{k-i,l-j}^{(G)},$
(3.1.22)

where

$\begin{gathered}f_{1,0}=3(6k-1),\quad f_{1,1}=2(6k-1),\quad
f_{2,0}=36(4-3k),\\\ f_{2,1}=48(4-3k),\quad
f_{2,2}=49k^{3}-108k^{2}+23k+40,\quad f_{3,0}=108(2k-5),\\\
f_{3,1}=216(2k-5),\quad f_{3,2}=3(5-2k)(98k^{2}-304k+189),\\\
f_{3,3}=2(5-2k)(98k^{2}-304k+221),\quad f_{4,2}=\tfrac{441}{2}(2k-5)_{3},\quad
f_{4,3}=294(2k-5)_{3},\\\
f_{4,4}=\tfrac{1}{2}(2k-5)_{3}\left(126k(3-k)-137\right),\quad
f_{5,4}=\tfrac{189}{2}(2k-5)_{5},\\\ f_{5,5}=63(2k-5)_{5},\quad
f_{6,6}=\tfrac{81}{8}(2k-5)_{7},\end{gathered}$

and all other $f_{i,j}$ are zero. We also set $M_{k,l}^{(G)}=0$ if $l<0$ or
$l>k$.

When equation (3.1.22) is used to compute $M_{k,0}^{(G)}$, it reduces to

$16(k+1)M_{k,0}^{(G)}=12(6k-1)(\kappa-1)^{2}M_{k-1,0}^{(G)}-36(3k-4)(\kappa-1)^{4}M_{k-2,0}^{(G)}\\\
+27(2k-5)(\kappa-1)^{6}M_{k-3,0}^{(G)},\quad\kappa=1/3\textrm{ or }3.$
(3.1.23)

This is in keeping with the limiting scaled eigenvalue density
$\rho^{(G),0}(\lambda)$ of the Gaussian ensembles equalling the Wigner semi-
circle law specified by the density $\sqrt{2-\lambda^{2}}/\pi$ (1.2.14)
supported on $|\lambda|<\sqrt{2}$: up to a scale factor the Catalan numbers
are the even moments. Thus, equation (3.1.23) can essentially be seen to be a
four term linear recurrence for the Catalan numbers. To the best of the
author’s knowledge, it does not have a combinatorial interpretation comparable
to what is known for the traditional two term recurrence given in equation
(3.1.25) below. However, we note that equation (3.1.23) and its three term
analogue (3.1.27) are implied by repeated applications of equation (3.1.25)
with appropriate scaling.

To expand the Laguerre and Jacobi ensembles’ spectral moments in $N$, we need
to decide how the parameters $a$ and $b$ scale with $N$. We were faced with
this same decision in §2.4.1, where we opted to set $a=\hat{a}\kappa N$ and
$b=\hat{b}\kappa N$ with $\hat{a},\hat{b}$ constant in $N$. We continue with
this choice for consistency, with one caveat: Since the $1$-point recursion in
the LUE case is relatively simple, we consider the slightly more general
parametrisation $a=\hat{a}\kappa N+\delta_{a}$, where $\hat{a},\delta_{a}={\rm
O}(1)$ — this gives us an opportunity to demonstrate the insight gained, at
the cost of complexity, from implementing this generalisation. Substituting
this generalised parametrisation together with the expansion (3.0.9) into
recurrence (3.1.16) yields the LUE $1$-point recursion.

######  Proposition 3.5.

For $\beta=2$, $a=\hat{a}N+\delta_{a}$ with $\hat{a},\delta_{a}$ constant in
$N$, and $k\geqslant 2$,

$(k+1)M_{k,l}^{(LUE)}=(2k-1)\left[(2+\hat{a})M_{k-1,l}^{(LUE)}+\delta_{a}M_{k-1,l-1}^{(LUE)}\right]\\\
-\hat{a}^{2}(k-2)M_{k-2,l}^{(LUE)}+(k-2)\left((k-1)^{2}-\delta_{a}^{2}\right)M_{k-2,l-2}^{(LUE)},$
(3.1.24)

where we set $M_{k,l}^{(LUE)}=0$ if $l<0$ or $l>k$.

When this is used to compute $M_{k,0}^{(LUE)}$ in the case $a=\delta_{a}={\rm
O}(1)$ and thus $\hat{a}=0$, one recovers the familiar Catalan recursion

$(k+1)M_{k,0}^{(LUE)}=2(2k-1)M_{k-1,0}^{(LUE)}.$ (3.1.25)

This recurrence is well known (see, e.g., [290]) and can be seen to be
consistent with Proposition 1.2: When $a={\rm O}(1)$, we have from equation
(1.2.88) that $\gamma_{1}=1$ so that the Marčenko–Pastur density
$\rho^{(L),0}(\lambda)$ (1.2.15) simplifies to
$\sqrt{4/\lambda-1}/(2\pi)\chi_{0<\lambda<4}$, whose $k\textsuperscript{th}$
spectral moment is exactly the $k\textsuperscript{th}$ Catalan number. Observe
also that the Harer–Zagier recursion (3.0.12) reduces to (3.1.25) upon setting
$l=0$ and noting that $M_{k,0}^{(LUE)}=2^{k}M_{k,0}^{(GUE)}$, this equality
being implied by the fact that
$\rho^{(G),0}(\lambda)=2|\lambda|\,\rho^{(L),0}(2\lambda^{2})$.

In contrast to the four term recurrence (3.1.23), equation (3.1.25) has a
famous combinatorial interpretation, which we now present in the language of
Section 3.3. For $k\in\mathbb{N}$ and $\beta=2$, the expansion coefficients
$M_{k,0}^{(LUE)}$ count the number of unique planar ribbon graphs that can be
constructed from a $2k$-gon111In §3.3.2, we will see that the ribbon graphs
pertaining to the Laguerre ensembles must obey a certain bicolouring
constraint. This constraint is automatically satisfied when the ribbon graphs
are planar, so we ignore it for now. [82]. To see that they are equal to the
$k\textsuperscript{th}$ Catalan number, observe that the set of planar ribbon
graphs that can be constructed from a $2k$-gon are in bijection with the set
of balanced parenthesisations involving $k$ pairs of parentheses, which are
well known to be enumerated by the Catalan numbers [290]; we make the
bijection explicit by moving clockwise around the $2k$-gon, starting at the
top edge, assigning the label ‘$1$’ (open parenthesis) to the first free edge
and ‘$-1$’ (corresponding closed parenthesis) to the edge connected to it by a
ribbon, ‘$2$’ and ‘$-2$’ to the next free edge and its ribbon-connected
counterpart, and so on. As exemplified in Figure 3.2 below, the edges are then
labelled — in clockwise order starting at the top edge — by the sequence
$(i_{1},i_{2},\ldots,i_{2k})$ with $i_{1}=1$ and $j<l$ whenever
$0<i_{j}<i_{l}$ or $0<i_{j}=-i_{l}$.

Figure 3.2: The first image relates a planar ribbon graph to the balanced
parenthesisation $()(())$ by labelling the top edge ‘$1$’ and then, moving
clockwise, labelling the tail of the ribbon connected to the top edge ‘$-1$’,
the head of the next ribbon ‘$2$’, the head of the final ribbon ‘$3$’, and
then the tails of these ribbons ‘$-3$’ and ‘$-2$’. In the second illustration,
we highlight the edges of the hexagon which fit the edge-marking criteria
below; in addition to the sixth edge in the clockwise ordering, which is
special, an edge is eligible if it precedes an edge that is the tail of some
ribbon. In the final image, we show how to split the vertex preceding the top
edge in the clockwise ordering so that there are seven vertices available for
the vertex-marking procedure outlined below.

To elucidate the combinatorics underlying equation (3.1.25), we take the
ribbon graphs labelled in the manner described above and assign them markings
in one of two ways. The first marking convention requires that one edge of the
$2k$-gon be marked, with eligible edges being those labelled by $i_{j}$ with
either $j=2k$ or $j$ such that $i_{j+1}<0$; that is, we mark either the final
edge in the clockwise ordering, or one that immediately precedes a negatively-
labelled edge (see the second illustration of Figure 3.2 above). There are
$(k+1)$ such eligible edges, so that the set of $k$-ribbon graphs edge-marked
in this fashion are enumerated by the left-hand side of equation (3.1.25). For
the second marking convention, we split the vertex connecting the first and
last edges so that, in clockwise ordering, we have a vertex immediately
preceding the first edge, a vertex immediately following the last edge, and
$2k-1$ vertices in between. Labelling one of these $2k+1$ vertices by either
of $\pm 0$ results in a set of $2(2k+1)M_{k,0}^{(LUE)}$ vertex-marked ribbon
graphs. Since this number is just the right-hand side of equation (3.1.25)
with $k$ replaced by $k+1$, it is unsurprising that there is a bijection
between the set of $k$-ribbon graphs edge-marked in the first described
convention and the set of $(k-1)$-ribbon graphs vertex-marked in the second
convention. To map from the first set to the second, delete the marked edge,
the ribbon connected to it, and the edge at the other end of said ribbon, so
that each deleted edge collapses to a vertex. Of these two edge-turned-
vertices, mark the one corresponding to the deleted edge that was positively-
labelled; label it ‘$-0$’ if the positively-labelled edge was marked, or ‘$0$’
otherwise. Finally, we split the vertex connecting the first and last edges.
For the reverse mapping, split the marked vertex into two and connect them
with an edge. If the vertex was marked with a ‘$-0$’, insert a marked edge
immediately anticlockwise to the new edge, and connect the two with a ribbon.
If the vertex was instead marked with a ‘$0$’, we need to travel clockwise to
a second vertex, turn it into a marked edge, and connect said edge to the
first through a ribbon; to uniquely determine the second vertex, we simply
pick the furthest one in the clockwise ordering (stopping before reaching the
first vertex and edge) that allows us to perform this procedure whilst still
producing a planar ribbon graph.

######  Example 3.1.

The planar ribbon graphs displayed in Figure 3.3 below map to each other via
the bijection detailed above. On the left, we have a planar $3$-ribbon graph
labelled by the sequence $(1,-1,2,3,-3,-2)$, with a marking on the final edge
‘$-2$’. On the right, we have a planar $2$-ribbon graph labelled by the
sequence $(1,-1,2,-2)$, with the vertex between edges ‘$-1$’ and ‘$2$’ marked
with a ‘$+0$’. To map the left image to the right, we delete the marked edge,
together with its partner ‘$2$’ and the ribbon connecting them, then relabel
the edges $3,-3$ as $2,-2$ and split the vertex connecting edges ‘$1$’ and
‘$-2$’. Since the original edge ‘$2$’ (we focus on the positively-labelled
deleted edge) was nestled between edges ‘$-1$’ and ‘$3$’ and was unmarked, we
mark the corresponding vertex with a ‘$+0$’ (if the edge marking in the left
image was at the edge ’$2$’, this vertex-marking would have been a ‘$-0$’). To
obtain the left illustration from the right, we expand the vertex labelled
‘$+0$’ into an edge, the vertex $d$ into a marked edge, and connect the two
with a ribbon. Then, we join up the new marked edge and the top edge ‘$1$’ and
relabel the edges in the appropriate manner. The new marked edge has to be at
vertex $d$ because it is the furthest along in the clockwise ordering since
vertices $a,b$ appear earlier than vertex ‘$+0$’ in said ordering; note that
picking vertex $c$ would result in a non-planar (genus one) ribbon graph.

Figure 3.3: Representatives of the sets of edge-marked planar $3$-ribbon
graphs with a hexagon at their core and vertex-marked planar $2$-ribbon graphs
stemming from a square. These sets are in bijection and have cardinality given
manifestly by, respectively, the left- and right-hand sides of equation
(3.1.25) with $k=3$.

When we remove the constraint $l=0$, $M_{k,l}^{(LUE)}$ retains a combinatorial
interpretation if we fix $a=\delta_{a}=0$. Then, as discussed in §3.3.2, it
enumerates ribbon graphs that are now bicoloured and of genus $l/2$ [82] (when
$a=0$, $M_{k,l}^{(LUE)}$ vanishes for all odd $l$). Equivalently,
$M_{k,l}^{(LUE)}|_{a=0}$ counts the number of ways of forming a compact
orientable genus $l/2$ surface by gluing together the edges of a $2k$-gon
whose vertices are alternately coloured black and red, with the gluing
respecting this bicolouring. When $a=0$, the general-$l$ recursion (3.1.24)
reduces to a recursion that runs over even $l$, which was recently derived in
[86, Thrm. 4.1] as a means of enumerating the aforementioned gluings. Keeping
$\delta_{a}=0$, but letting $\hat{a}\geqslant 0$ be free,
$M_{k,l}^{(LUE)}|_{a=\hat{a}N}$ counts the same ribbon graphs, with the red
vertices now weighted by $(\hat{a}+1)$. Thus, $M_{k,l}^{(LUE)}|_{a=\hat{a}N}$
is a polynomial in $(\hat{a}+1)$ with the coefficient of $(\hat{a}+1)^{p}$
equalling the number of genus $l/2$ bicoloured gluings of the aforementioned
$2k$-gon that have $p$ distinct red vertices surviving the gluing procedure
(again, we refer to §3.3.2 for details). In this $a=\hat{a}N$ setting,
recursion (3.1.24) continues to run over even $l$ so that
$m_{k}^{(LUE)}|_{a=\hat{a}N}$ is manifestly an odd function in $N$, in keeping
with the second point of Remark 2.1.

A curious point to note is that even though the moment expansion coefficients
$M_{k,l}^{(LUE)}$ have combinatorial interpretations for $k,l\in\mathbb{N}$
and $a=\hat{a}N$ with $\hat{a}\geqslant 0$, the recursion (3.1.24) relating
them does not presently have such an interpretation (at least not one
comparable to the bijection demonstrated in Example 3.1 above). Indeed, it may
be the case that this $1$-point recursion describes a relation between the
$M_{k,l}^{(LUE)}|_{a=\hat{a}N}$ which simply does not correspond to a
topological procedure at the level of the ribbon graphs themselves —
ascertaining the validity of this intriguing proposition remains an open
problem. Similar to the LUE case, the moment expansion coefficients $M_{k,l}$
of the GUE, GOE, GSE, LOE, and LSE also have combinatorial meaning, to be
detailed in Section 3.3, but the $1$-point recursions characterising them are
yet to be understood in a combinatorial sense. With the GUE, GOE, and GSE
$1$-point recursions known from the works [174], [224], we now supply the
analogous recursions for the LOE and LSE.

######  Proposition 3.6.

For $\beta=1$ and $4$, $a=\hat{a}\kappa N$ with $\hat{a}$ constant in $N$, and
$k\geqslant 4$,

$(k+1)M_{k,l}^{(L)}=\sum_{i=1}^{4}\sum_{j=0}^{i}(\kappa-1)^{j}g_{i,j}M_{k-i,l-j}^{(L)},$
(3.1.26)

where

$\begin{gathered}g_{1,0}=(4k-1)\left(\hat{a}\kappa+4(\kappa-1)\right)(\kappa-1),\\\
g_{2,0}=(3-2k)\left(\hat{a}^{2}\kappa^{2}+2(\hat{a}\kappa+4(\kappa-1))^{2}(\kappa-1)^{2}\right),\quad
g_{2,1}=2(2k-3)\hat{a}\kappa,\\\ g_{2,2}=(k-1)\left(5k^{2}-11k+4\right),\quad
g_{3,0}=(4k-11)(\hat{a}\kappa+4(\kappa-1))\hat{a}^{2}\kappa^{2}(\kappa-1),\\\
g_{3,1}=2(11-4k)(\hat{a}\kappa+4(\kappa-1))\hat{a}\kappa(\kappa-1),\\\
g_{3,2}=2(3-k)\left(5k^{2}-19k+16\right)(\hat{a}\kappa+4(\kappa-1))(\kappa-1),\quad
g_{4,0}=(4-k)\hat{a}^{4}\kappa^{4},\\\
g_{4,1}=4(k-4)\hat{a}^{3}\kappa^{3},\quad
g_{4,2}=(k-4)\left(5k^{2}-32k+47\right)\hat{a}^{2}\kappa^{2},\\\
g_{4,3}=2(4-k)(k-3)(5k-17)\hat{a}\kappa,\quad
g_{4,4}=4(1-k)\left[(k-4)(k-3)\right]^{2},\end{gathered}$

and all other $g_{i,j}$ are zero. We also set $M_{k,l}^{(L)}=0$ if $l<0$ or
$l>k$.

When $a=0$, $M_{k,l}^{(LOE)}$ is shown in §3.3.2 to count the same bicoloured
ribbon graphs as $M_{k,l}^{(LUE)}$ except that the ribbons are now allowed to
have Möbius half-twists so that the ribbon graphs are no longer guaranteed to
be globally orientable, though they still have Euler genus $l$ (recall
Definition 3.1 and see Figures 3.15 and 3.19 for examples). Thus, in keeping
with the discussion comparing equations (3.1.23) and (3.1.25), we see that the
three term analogue of these equations obtained by setting $a=l=0$ in
Proposition 3.6,

$(k+1)M_{k,0}^{(L)}=4(4k-1)(\kappa-1)^{2}M_{k-1,0}^{(L)}-32(2k-3)(\kappa-1)^{4}M_{k-2,0}^{(L)},$
(3.1.27)

is, up to scaling, satisfied by the Catalan numbers when $\kappa=1/2$ or $2$.
Indeed, this recursion can be used to verify that
$2^{k}M_{k,0}^{(LOE)}|_{a=0}$, equivalently $2^{-k}M_{k,0}^{(LSE)}|_{a=0}$, is
equal to the $k\textsuperscript{th}$ Catalan number. Moreover, we observe that
equation (3.1.27) is equivalent to that obtained by setting $l=0$ in the GOE
$1$-point recursion given in [224, Cor. 7].

Now returning to the general-$l$ case, let us compare the moment expansion
coefficients $M_{k,l}^{(L)}$ and $M_{k,l}^{(G)}$. We immediately observe that
the former is somehow more complicated than the latter: $M_{k,l}^{(L)}$
enumerates the same ribbon graphs as $M_{k,l}^{(G)}$ but with the extra
bicolouring constraint described earlier and in §3.3.2, so that
$M_{k,l}^{(L)}|_{a=0}\leqslant M_{k,l}^{(G)}$ and when $a=\hat{a}\kappa N\neq
0$, $(\hat{a}+1)$ acts as a weight that keeps track of the ratio of red to
black vertices in the ribbon graphs. Note also that the presence of the
parameter $a$ results in the expansion coefficients $M_{k,l}^{(L)}$ and the
$1$-point recursions (3.1.24), (3.1.26) being more complex than their Gaussian
counterparts. This increase in complexity is linked to the fact that the
Laguerre ensembles sit above the Gaussian ensembles in the hierarchy implied
by Lemma 1.1. We posit that since the Jacobi ensembles inhabit the tier
immediately above the Laguerre ensembles in the aforementioned hierarchy, it
is not unreasonable to expect that the $M_{k,l}^{(J)}$ (3.0.10) count some
type of ribbon graphs that include, as special cases, the ribbon graphs