enumerated by the $M_{k,l}^{(G)}$ or $M_{k,l}^{(L)}$. However, there is no
such construction currently known in the literature, with attempts at
replicating the constructions of Section 3.3 thwarted by the fact that the
Jacobi matrices of Definition 1.4 involve inverse matrices whose entries are
not compatible with the Isserlis–Wick theorem (see the discussion following
Theorem 3.1 of Section 3.3). It is nonetheless the author’s belief, which we
justify below (see also the discussion in §4.4.3 regarding the works [326],
[25], [73], [159], [160]), that the $M_{k,l}^{(J)}$ should have combinatorial
interpretations in line with what is known for the Gaussian and Laguerre
ensembles.

To motivate the study of the Jacobi ensembles’ moment expansion coefficients
$M_{k,l}^{(J)}$ with a combinatorial flavour, let us review some properties of
the classical matrix ensembles that are relevant in the combinatorial setting.
To begin, recall from the paragraph preceding Corollary 3.2 that
$m_{k}^{(J)}$, consequently $M_{k,l}^{(J)}$, reduces to $m_{k}^{(L)}$,
respectively $M_{k,l}^{(L)}$, when taking $b\to\infty$ (possibly by setting
$b=\hat{b}\kappa N$ and working in the large $N$ limit), so that if
$M_{k,l}^{(J)}$ were to enumerate a class of ribbon graphs, these ribbon
graphs would degenerate to those enumerated by $M_{k,l}^{(L)}$ in the large
$b$ limit. This is understood to be a simple consequence of the limit (1.2.23)
presented in Lemma 1.1, which is key to comparing the Laguerre and Jacobi
ensembles. The other limit given in Lemma 1.1, relating the Jacobi and
Gaussian ensembles, has a cleaner form when written in terms of the symmetric
shifted Jacobi ensemble:

$\lim_{a\to\infty}w^{(sJ)}(\lambda/\sqrt{a})\big{|}_{a=b}=w^{(G)}(\lambda).$

A similar observation can be made by comparing the identities

$\displaystyle\rho^{(G),0}(\lambda)$
$\displaystyle=\sqrt{2}(1-\lambda^{2}/2)\rho^{(sJ),0}(\lambda/\sqrt{2})\big{|}_{a=b=0},$
(3.1.28) $\displaystyle\rho^{(G),0}(\lambda)$
$\displaystyle=|\lambda|(1-\lambda^{2}/2)\rho^{(J),0}(\lambda^{2}/2)\big{|}_{a=b=0}.$
(3.1.29)

Recalling the discussion following equation (3.1.25), equation (3.1.29) is
reminiscent of the analogous relation between $\rho^{(G),0}(\lambda)$ and
$\rho^{(L),0}(\lambda)$, suggesting that while the GUE can be compared at
leading order to the sJUE by a linear change of variables, the LUE is more
closely related to the JUE. If we expand $\mu_{k}^{(sJ)}$ (3.1.5) as

$\mu_{k}^{(sJ)}=\sum_{l=0}^{\infty}\Delta M_{k,l}^{(sJ)}N^{1-l}$ (3.1.30)

and define $\Delta M_{k,l}^{(J)}:=M_{k,l}^{(J)}-M_{k+1,l}^{(J)}$, equations
(3.1.28) and (3.1.29) imply that

$2^{k}\Delta M_{k,0}^{(J)}\big{|}_{a=b=0}=-2^{k+1}\Delta
M_{k,0}^{(sJ)}\big{|}_{a=0}=M_{k,0}^{(G)},$ (3.1.31)

which links the differences of the moments of $\rho^{(J),0}(\lambda)|_{a=b=0}$
and $\rho^{(sJ),0}(\lambda)|_{a=b=0}$ to the Catalan numbers. Combining these
observations together, it seems that there is merit in studying the moment
expansion coefficients of both the un-shifted and symmetric shifted Jacobi
ensembles. At this point in time, a good approach for progressing this line of
study is to find more ways of comparing the moment expansion coefficients
$M_{k,l}$ of the classical matrix ensembles. To this end, we now derive the
$1$-point recursions satisfied by $M_{k,l}^{(JUE)}$, $\Delta M_{k,l}^{(JUE)}$,
and $\Delta M_{k,l}^{(sJUE)}$, fixing $\beta=2$ for clarity.

######  Proposition 3.7.

Fix $\beta=2$, $a=\hat{a}N$, and $b=\hat{b}N$, with $\hat{a},\hat{b}={\rm
O}(1)$. Then, for $k\geqslant 3$,

$k(\hat{a}+\hat{b}+2)^{2}M_{k,l}^{(JUE)}=k(k-1)^{2}M_{k,l-2}^{(JUE)}+\sum_{i=1}^{3}\left(h_{i,0}M_{k-i,l}^{(JUE)}+h_{i,1}M_{k-i,l-2}^{(JUE)}\right),$
(3.1.32)

where

$\begin{gathered}h_{1,0}=2(4k-3)(\hat{a}+\hat{b}+1)+\left(3\hat{a}(k-1)+\hat{b}k\right)(\hat{a}+\hat{b}),\\\
h_{1,1}=(1-k)(3k^{2}-8k+6),\\\
h_{2,0}=3\hat{a}^{2}(2-k)+(3-2k)\left((\hat{a}+2)(\hat{b}+2)-2\right),\\\
h_{2,1}=(k-2)(3k^{2}-10k+9),\quad h_{3,0}=\hat{a}^{2}(k-3),\quad
h_{3,1}=(3-k)(k-2)^{2},\end{gathered}$

and we set $M_{k,l}^{(JUE)}=0$ if $l<0$.

###### Proof.

The derivation of this recursion is slightly different to that described
earlier, so we supply the details. Substitute expansion (3.0.10) into the
recurrence (3.1.3), along with the substitutions $a=\hat{a}N$ and
$b=\hat{b}N$, and then equate terms of order $3-l$ in $N$. Note that with our
choice of $a$ and $b$, the coefficients $d_{2,0}^{(J)},\ldots,d_{2,3}^{(J)}$
all contain a term of order two and a term of order one in $N$, and no other
terms. The ${\rm O}(N^{2})$ terms of $d_{2,i}^{(J)}$ correspond to the
coefficients $h_{i,0}$ of $M_{k-i,l}^{(JUE)}$ in equation (3.1.32), while the
terms of order one in $N$ correspond to the coefficients $h_{i,1}$ of
$M_{k-i,l-2}^{(JUE)}$. ∎

Proposition 3.7 leads directly to the $1$-point recursion on the $\Delta
M_{k,l}^{(JUE)}|_{a=\hat{a}N,\,b=\hat{b}N}=M_{k,l}^{(JUE)}|_{a=\hat{a}N,\,b=\hat{b}N}-M_{k+1,l}^{(JUE)}|_{a=\hat{a}N,\,b=\hat{b}N}$,
while applying the inverse of equation (2.2.3) yields the equivalent relation
on the $\Delta M_{k,l}^{(sJUE)}|_{a=\hat{a}N}$ (3.1.30). Alternatively, the
latter can be derived from equation (3.1.7) by following the prescription at
the beginning of this subsection.

######  Corollary 3.3.

In the setting of Proposition 3.7 with $\Delta
M_{k,l}^{(JUE)}=M_{k,l}^{(JUE)}-M_{k+1,l}^{(JUE)}$ and $\Delta
M_{k,l}^{(sJUE)}$ defined implicitly by equation (3.1.30), we have for
$k\geqslant 2$ the $1$-point recursions

$(k+1)(\hat{a}+\hat{b}+2)^{2}\Delta
M_{k,l}^{(JUE)}=(2k-1)\left(\hat{a}(\hat{a}+\hat{b}+2)+2(\hat{b}+1)\right)\Delta
M_{k-1,l}^{(JUE)}\\\ +(2-k)\hat{a}^{2}\Delta
M_{k-2,l}^{(JUE)}+k^{2}(k+1)\Delta M_{k,l-2}^{(JUE)}\\\ -k(k-1)(2k-1)\Delta
M_{k-1,l-2}^{(JUE)}+(k-2)(k-1)^{2}\Delta M_{k-2,l-2}^{(JUE)}$ (3.1.33)

and

$4(k+1)(\hat{a}+1)^{2}\Delta M_{k,l}^{(sJUE)}=2(2k-1)(2\hat{a}+1)\Delta
M_{k-1,l}^{(sJUE)}+(k+1)(2k+1)^{2}\Delta M_{k,l-2}^{(sJUE)}\\\
+4k^{2}(2k-1)\Delta M_{k-1,l-2}^{(sJUE)}+(k-1)(2k-1)(2k-3)\Delta
M_{k-2,l-2}^{(sJUE)},$ (3.1.34)

with $\Delta M_{k,l}^{(JUE)},\Delta M_{k,l}^{(sJUE)}$ set to zero for all
$l<0$.

There are three immediate observations relating to Proposition 3.7 and
Corollary 3.3. Firstly, the recursions therein run over even $l$, similar to
Proposition 3.5. Indeed, when the first few moments
$m_{k}^{(JUE)},m_{k}^{(sJUE)}$ with $k=0,\ldots,3$ (known from [96], [242],
[142]) are expanded as series in $1/N$, they do not contain terms of even
powers in $N$, so the recursions (3.1.32)–(3.1.34) are trivially satisfied
when $l$ is odd. As with Proposition 3.5, this feature is due to our choice of
taking $\beta=2$ and $a,b\propto N$. Our second observation is that, unlike
the analogous $1$-point recursions for the GUE and LUE moment expansion
coefficients, the recursions of Proposition 3.7 and Corollary 3.3 show that
$M_{k,l}^{(JUE)},\Delta M_{k,l}^{(JUE)},\Delta M_{k,l}^{(sJUE)}$ depend,
respectively, on $M_{k,l-2}^{(JUE)},\Delta M_{k,l-2}^{(JUE)},\Delta
M_{k,l-2}^{(sJUE)}$. Thus, in contrast to the Gaussian and Laguerre cases,
knowledge of $m_{k^{\prime}}^{(JUE)},\Delta
m_{k^{\prime}}^{(JUE)},\mu_{k^{\prime}}^{(sJUE)}$ with $k^{\prime}<k$ is not
enough to determine $M_{k,l}^{(JUE)},\Delta M_{k,l}^{(JUE)},\Delta
M_{k,l}^{(sJUE)}$ for $l>0$. Our final observation is that of the three
recursions (3.1.32)–(3.1.34), the final one is the simplest.

If we are to assume that the moment expansion coefficients of the Jacobi
ensembles count some type of surface similar to those counted by the GUE and
LUE spectral moments, comparing the corresponding $1$-point recursions may
shed light on what these surfaces actually are. For example, the fact that the
recursions of Proposition 3.7 and Corollary 3.3 run over even $l$ suggests
that $l/2$ might play the role of genus in the Jacobi case, as it does in the
Gaussian and Laguerre cases. On the other hand, the second observation given
above could give a hint as to what sets the JUE apart from the GUE and LUE.
Since setting $a=0$ in Proposition 3.5 yields a simpler recursion that retains
an interpretation in terms of ribbon graphs, we set $a=b=0$ in Proposition 3.7
and Corollary 3.3 for comparison:

######  Corollary 3.4.

In the context of Proposition 3.7 and Corollary 3.3, setting $a=b=0$ reduces
recursion (3.1.32) to

$4kM_{k,l}^{(JUE)}=k(k-1)^{2}M_{k,l-2}^{(JUE)}+2(4k-3)M_{k-1,l}^{(JUE)}\\\
+(1-k)(3k^{2}-8k+6)M_{k-1,l-2}^{(JUE)}+2(3-2k)M_{k-2,l}^{(JUE)}\\\
+(k-2)(3k^{2}-10k+9)M_{k-2,l-2}^{(JUE)}+(3-k)(k-2)^{2}M_{k-3,l-2}^{(JUE)},$
(3.1.35)

recursion (3.1.33) to

$4(k+1)\Delta M_{k,l}^{(JUE)}=2(2k-1)\Delta M_{k-1,l}^{(JUE)}+k^{2}(k+1)\Delta
M_{k,l-2}^{(JUE)}\\\ -k(k-1)(2k-1)\Delta
M_{k-1,l-2}^{(JUE)}+(k-2)(k-1)^{2}\Delta M_{k-2,l-2}^{(JUE)},$ (3.1.36)

and recursion (3.1.34) to

$4(k+1)\Delta M_{k,l}^{(sJUE)}=2(2k-1)\Delta
M_{k-1,l}^{(sJUE)}+(k+1)(2k+1)^{2}\Delta M_{k,l-2}^{(sJUE)}\\\
+4k^{2}(2k-1)\Delta M_{k-1,l-2}^{(sJUE)}+(k-1)(2k-1)(2k-3)\Delta
M_{k-2,l-2}^{(sJUE)}.$ (3.1.37)

Comparing these forms to the GUE analogue (3.0.12) and the result [86, Thrm.
4.1] of setting $a=0$ in Proposition 3.5,

$(k+1)M_{k,l}^{(LUE)}=2(2k-1)M_{k-1,l}^{(LUE)}+(k-2)(k-1)^{2}M_{k-2,l-2}^{(LUE)},$
(3.1.38)

we note that equations (3.1.36) and (3.1.37) contain familiar terms, whereas
the same is not true for equation (3.1.35). Setting equation (3.1.35) aside,
we see that upon ignoring the terms with indices $(k,l-2)$ and $(k-1,l-2)$,
equation (3.1.36) has the same structure as equation (3.1.38), while equation
(3.1.37) is almost equivalent to the Harer–Zagier recursion (3.0.12). Hence,
it seems that the JUE should be compared to the LUE and the sJUE should be
compared to the GUE (cf. the discussion following equations (3.1.28) and
(3.1.29)). In addition, the coefficients of the ignored terms contain,
respectively, the factors $(k+1)$ and $(2k-1)$, which suggests that we should
group them with the terms with indices $(k,l)$ and $(k-1,l)$. As expected,
setting $l=0$ in equations (3.1.36) and (3.1.37) shows that $2^{k}\Delta
M_{k,0}^{(JUE)}|_{a=b=0}$ and $-2^{k+1}\Delta M_{k,0}^{(sJUE)}|_{a=0}$ satisfy
the two term Catalan recursion (3.1.25), which is in keeping with equation
(3.1.31). Moving past these observations, the challenge now is to extend the
combinatorial interpretation of equation (3.1.25) to the general-$l$
recursions (3.0.12), (3.1.36), (3.1.37), and (3.1.38). As mentioned earlier,
this remains an open problem that we are unable to address here. Nonetheless,
some comments on promising directions are given in §4.4.2. As we move on to
the next section, let us make a final remark: Experimentation for $0\leqslant
k\leqslant 10$, $0\leqslant l\leqslant 5$ suggests that for each $k,l\geqslant
0$, $-2^{2(k+l)+1}\Delta M_{k,l}^{(sJUE)}|_{a=0}$ is a positive integer, which
supports the idea that these scaled differences of moments are equal to the
cardinality of some class of combinatorial objects. Moreover, it can be seen
that these positive integers grow extremely quickly as $k,l$ are increased,
relative to what is seen for their GUE and LUE counterparts; to be more
concrete, comparing equations (3.0.12) and (3.1.37) shows that for all
$k,l\geqslant 0$,

$-2^{2(k+l)+1}\Delta M_{k,l}^{(sJUE)}|_{a=0}\geqslant M_{k,l}^{(GUE)}\geqslant
M_{k,l}^{(LUE)}|_{a=0}.$

###  3.2 Established Results on the Moments of the Classical Matrix Ensembles

In the previous section, we derived recursions characterising the spectral
moments of the classical matrix ensembles, along with their complex-$k$
generalisations (3.1.1) and their expansion coefficients $M_{k,l}$. To
summarise Sections 2.3 and 3.1, our methodology consisted of using the theory
of Selberg correlation integrals to obtain differential equations satisfied by
the eigenvalue densities $\rho(x)$ and resolvents $W_{1}(x)$ of the classical
matrix ensembles — applying integration by parts on the former set of
differential equations or inserting the expansion
$W_{1}(x)=\sum_{k=0}^{\infty}m_{k}/x^{k+1}$ in the latter set resulted in the
sought recursions. Some of the recursions derived in Section 3.1 have already
been given in the literature, but our approach involving Selberg correlation
integrals is distinct and is able to treat all of the classical matrix
ensembles in a unified manner. In this section, we give a select review on
recent studies that provide characterisations of the moments of the classical
matrix ensembles, focusing particularly on those that are different to the
results of Section 3.1 (classical results on this topic are reviewed in a
plethora of texts, including [238], [119]). Seeing as this gives us an
opportunity to showcase connections to various fields of mathematics, we
compartmentalise our review by methodology rather than outcome.

####  3.2.1 Results from skew-orthogonal polynomial theory

A straightforward approach to computing the spectral moments is to simply
integrate monomials against closed form expressions for the corresponding
eigenvalue densities. This approach was taken by Livan and Vivo in 2011 [229],
with their starting point being expressions for the classical Laguerre and
Jacobi ensembles’ eigenvalue densities known from (skew-)orthogonal polynomial
theory. Recalling the discussion of §1.2.3, the idea then is to take
$\rho(\lambda)$ to be given by equation (1.2.67) with $x=y=\lambda$ when
$\beta=2$, and by this plus a correction term when $\beta=1,4$. Integrating
these expressions against $\lambda^{k}$ for $k\in\mathbb{N}$, Livan and Vivo
gave exact expressions for the moments of the classical Laguerre and Jacobi
ensembles in terms of sums of ratios of gamma functions. At this same time, in
a parallel but independent work, Mezzadri and Simm [240] derived equivalent
expressions for the spectral moments $m_{k}$ of the classical Gaussian,
Laguerre, and Jacobi ensembles, moreover showing that their expressions hold
true for all $k\in\mathbb{C}$ such that the $m_{k}$ are well-defined. In order
to ease the required integrations against the aforementioned expressions for
the eigenvalue densities $\rho^{(w)}(\lambda)$, a key idea of Mezzadri and
Simm was to use polynomials that are orthogonal with respect to the
perturbations $\lambda^{k}w(\lambda)$ of the classical weights $w(\lambda)$
(1.2.9).

The moment recurrences given in [169], [223], [224], [71] were also obtained
through consideration of (skew-)orthogonal polynomial theory. In the 2003 work
[169], Haagerup and Thorbjørnsen used elementary identities satisfied by the
Hermite and Laguerre orthogonal polynomials to express the exponential moment
generating function (cf. equation (1.1.18))

$R_{1}(x):=\int_{\mathrm{supp}\,\rho}\exp(x\lambda)\rho(\lambda)\,\mathrm{d}\lambda,$

for the GUE and LUE cases in terms of (confluent) hypergeometric functions.
They then transformed known differential equations satisfied by these
hypergeometric functions into differential equations characterising
$R_{1}(x)$. From there, Haagerup and Thorbjørnsen obtained recurrences for the
GUE and LUE (positive-integer) spectral moments through essentially the same
strategy as used in §3.1.1, thereby recovering the recursion of Harer and
Zagier [174], and finding its LUE analogue. Ledoux [223] extended these ideas
to the JUE case a year later, and to the GOE case [224] in 2009 — clever
manipulations were needed to handle the correction term that must be added to
the GUE eigenvalue density to obtain its GOE analogue. In 2016, Cunden et al.
[71] extended the work of Ledoux to derive an inhomogeneous moment recurrence
for the LOE, where the inhomogeneous terms were given in terms of the LUE
moments. This recurrence, together with the LUE moment recurrence given in
[169], was shown to hold for negative-integer moments ($m_{-k}$ with
$0<k<a+1$).

####  3.2.2 Results from symmetric function theory

Of the classical $\beta$ ensembles (recall §1.2.4), only the $\beta=1,2$, and
$4$ regimes can be studied through (skew-)orthogonal polynomial theory. To
compute the spectral moments in the general $\beta$ case, other approaches
must be taken. One option is to use loop equation analysis to iteratively
compute the resolvent expansion coefficients
$W_{1}^{0}(x),W_{1}^{1}(x),\ldots$ (1.1.21) and then interpret them as
generating functions for the moment expansion coefficients $M_{k,l}$. In the
Gaussian and Laguerre cases, the spectral moments are polynomials in $N$, so
this procedure can be used to compute the spectral moment $m_{k}$ for any
given $k$, assuming enough computation power is available; in the case of the
Jacobi and Cauchy $\beta$ ensembles, the best one can hope for is a truncation
of the large $N$ expansion of the spectral moments. We do not review the loop
equation formalism any further here since it will be revisited in Chapter 4,
but note that the Gaussian, Laguerre, and Jacobi $\beta$ ensembles were
studied in this manner in the 2014 and 2017 works [319], [142].

While the loop equation formalism can be used to compute (the leading order
behaviour of) spectral moments of the classical $\beta$ ensembles, it does not
provide any closed form expressions for these moments. Such closed form
expressions can however be obtained in the general $\beta$ setting through
symmetric function theory. The relevance of said theory should not be
surprising since equation (3.0.1) shows that the $k\textsuperscript{th}$
spectral moment is given by the integral of the symmetric polynomial
$\sum_{i=1}^{N}\lambda_{i}^{k}$ multiplied by the appropriate eigenvalue
j.p.d.f. Some well known bases for the ring of symmetric polynomials are the
sets of monomial symmetric polynomials, Schur polynomials, zonal polynomials,
and the Jack polynomials. As an algebra, it is also generated by the sets of
power-sum symmetric polynomials and elementary symmetric polynomials. We refer
the reader to [232] for definitions of these polynomials, but note that the
elementary symmetric polynomials were mentioned earlier in Remark 2.2. Of
particular interest to us is the set of Jack polynomials of type “C”, which
differ from those of type “J” and “P” by a choice of normalisation (see, e.g.,
[92]) and are written as

$C_{\sigma}^{(\alpha)}(\lambda_{1},\ldots,\lambda_{N}).$

They are indexed by partitions $\sigma$ of integers (i.e.,
$\sigma=(\sigma_{1},\ldots,\sigma_{k})$ with
$k,\sigma_{1},\ldots,\sigma_{k}\in\mathbb{N}$ and
$\sigma_{1}\geqslant\sigma_{2}\geqslant\cdots\geqslant\sigma_{k}>0$) and a
positive real parameter $\alpha$ not to be confused with that associated with
the Cauchy ensemble; setting $\alpha=1$ recovers the Schur polynomials while
$\alpha=2$ corresponds to the zonal polynomials.

In the 1997 work on Selberg correlation integrals [199], Kadell gave an
explicit closed form expression, in terms of ratios of gamma functions, for
the integral

$\int_{[0,1]^{N}}C_{\sigma}^{(2/\beta)}(\lambda_{1},\ldots,\lambda_{N})\,p^{(J)}(\lambda_{1},\ldots,\lambda_{N};\beta)\,\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{N},$
(3.2.1)

where $p^{(J)}(\lambda_{1},\ldots,\lambda_{N};\beta)$ is the eigenvalue
j.p.d.f. (1.2.81) of the Jacobi $\beta$ ensemble and $\sigma$ is any given
partition. In the same year, Baker and Forrester [27] showed that when
$w(\lambda)$ is either the Gaussian or Laguerre weight (1.2.9), the above
integral (3.2.1) with $p^{(J)}$ replaced by $p^{(w)}$ is equal to the
generalised Hermite (up to a minus sign), respectively Laguerre, polynomial of
index $\sigma$ evaluated at zero, these generalised polynomials being the
multivariable symmetric polynomials that are orthogonal with respect to the
inner products

$\left<f,g\right>^{(w)}:=\int_{\mathbb{R}^{N}}f(\lambda_{1},\ldots,\lambda_{N})g(\lambda_{1},\ldots,\lambda_{N})\,p^{(w)}(\lambda_{1},\ldots,\lambda_{N};\beta)\,\mathrm{d}\lambda_{1}\cdots\mathrm{d}\lambda_{N}.$

In the Jacobi case, one can surmise from [27] a result analogous to that of
Kadell [199].

Around 2003 [92], Dumitriu et al. used the results of [199], [27] (among
others) to devise algorithms, implemented in the MOPS package [96], for
computing the spectral moments of the Gaussian, Laguerre, and Jacobi $\beta$
ensembles. These algorithms essentially boil down to expressing the polynomial
$\sum_{i=1}^{N}\lambda_{i}^{k}$ as a linear combination of Jack “C”
polynomials so that the problem of computing the spectral moments (3.0.1)
reduces to the computation of integrals of the form (3.2.1). One should note
that these algorithms are not equivalent to closed form expressions for the
spectral moments. However, in the 2017 work [242], Mezzadri et al. provide
such closed form expressions in the Jacobi and Laguerre cases by way of giving
explicit formulae for the coefficients in the aforementioned linear
combination of Jack “C” polynomials that is equal to
$\sum_{i=1}^{N}\lambda_{i}^{k}$. Thus, they express $m_{k}^{(J)}$ and
$m_{k}^{(L)}$ as sums over partitions of $k$ with each summand being an
explicit ratio of gamma functions. Prompted by Fyodorov and Le Doussal, they
also show how to use a relation between
$\left\\{C_{\sigma}^{(\alpha)}(1/\lambda_{1},\ldots,1/\lambda_{N})\right\\}_{\sigma}$
and
$\left\\{C_{\sigma^{\prime}}^{(\alpha)}(\lambda_{1},\ldots,\lambda_{N})\right\\}_{\sigma^{\prime}}$
to extend their expressions for $m_{k}^{(J)},m_{k}^{(L)}$ to the case that $k$
is a negative integer. In fact, these expressions for the integer moments of
the Jacobi and Laguerre $\beta$ ensembles as sums of ratios of gamma functions
were given by Fyodorov and Le Doussal in the independent work [149]. Moreover,
this latter work contains contour integral representations for
$m_{k}^{(J)},m_{k}^{(L)}$ ($k\in\mathbb{Z}$ sufficiently large) due to Borodin
and Gorin. Before moving on, let us finally mention that Novaes also studied
the negative-integer moments of the LUE using Schur polynomials in the 2015
work [267].

####  3.2.3 Results in terms of hypergeometric orthogonal polynomials

In the recent work [72], Cunden at al. made the novel observation that in the
classical Gaussian, Laguerre, and Jacobi cases, simple linear combinations of
the complex-$k$ moments $m_{k}$ (3.1.1), interpreted as functions of $k$ and
renormalised by factors dependent on $N$, are hypergeometric orthogonal
polynomials in $k$. To be precise, they showed that when $\beta=2$ and $N$ is
fixed, $m_{k}^{(G)}$ is essentially a Meixner–Pollaczek polynomial in $k$,
$m_{k}^{(L)}$ is a continuous dual Hahn polynomial, and $\Delta
m_{k}^{(J)}=m_{k}^{(J)}-m_{k+1}^{(J)}$ is a Wilson polynomial. These three
polynomials belong to the Askey scheme of hypergeometric orthogonal
polynomials, with the first two being degenerations of the latter. Cunden et
al. prove this observation in the Gaussian and Laguerre cases by seeing
agreement in the hypergeometric function representations of the moments and
the corresponding polynomials when $k\in\mathbb{N}$, and then extending this
agreement to $k\in\mathbb{C}$ through an application of Carlson’s theorem (cf.
Remark 2.5); in the Jacobi case, lacking such a representation in terms of
hypergeometric functions, they derive the analogue of recurrence (3.1.3) for
$\Delta m_{k}^{(J)}$ and show that this recurrence, after appropriate scaling,
also characterises the Wilson polynomials in a unique sense. By noting that
certain linear combinations of the $\beta=1,4$ moments are known to be given
by linear combinations of their $\beta=2$ counterparts, Cunden et al. further
show within [72] how the polynomial characterisations of the $\beta=2$ moments
extends to the linear combinations considered in the $\beta=1,4$ cases.

######  Remark 3.1.

It was shown in [72] that the LUE moments and JUE moment differences satisfy
the following reciprocity laws:

$\displaystyle m_{-k-1}^{(L)}$
$\displaystyle=\left(\prod_{j=-k}^{k}\frac{1}{a-j}\right)m_{k}^{(L)},$ (3.2.2)
$\displaystyle\Delta m_{-k-1}^{(J)}$
$\displaystyle=\left(\prod_{j=-k}^{k}\frac{a+b+2N-j}{a-j}\right)\Delta
m_{k}^{(J)}.$ (3.2.3)

It is not immediately obvious that there exist similar laws for the moments of
the orthogonal or symplectic ensembles; one can experiment using data
computable from Proposition 3.1 and Corollary 3.2. It is believed that this is
a consequence of three term recurrences, as seen in the $\beta=2$ cases, being
simpler than their five term ($\beta=1,4$) analogues in an integrable sense.
Indeed, this is the reason that Cunden et al. were able to place the $\beta=2$
moments $m_{k}^{(G)},m_{k}^{(L)},m_{k}^{(J)}$ in the Askey scheme of
hypergeometric orthogonal polynomials, but were only able to do so for linear
combinations of the moments when taking $\beta=1,4$.

Soon after the work [72], Assiotis et al. showed [23] that the complex-$k$ sum
of moments $\mu_{k}^{(CyUE)}$ (3.1.6) of the symmetric Cauchy unitary
ensemble, again understood as a function of $k$ with fixed $N$ and properly
renormalised, is also a hypergeometric orthogonal polynomial. This time, the
relevant polynomials are the continuous Hahn polynomials, which are also
degenerations of the Wilson polynomials. Recalling that the hypergeometric
functions are defined by

$\,{}_{p}F_{q}\bigg{(}\begin{matrix}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{matrix}\,\bigg{|}\,z\bigg{)}:=\sum_{n=0}^{\infty}\frac{(a_{1})^{(n)}\cdots(a_{p})^{(n)}}{(b_{1})^{(n)}\cdots(b_{q})^{(n)}}\frac{z^{n}}{n!},$

with $(x)^{(n)}=x(x+1)\cdots(x+n-1)$ denoting the rising Pochhammer symbol,
the continuous Hahn polynomials are defined as [207]

$S_{n}(x;a,b,c,d):=\mathrm{i}^{n}\frac{(a+c)^{(n)}(a+d)^{(n)}}{n!}\,{}_{3}F_{2}\bigg{(}\begin{matrix}-n,n+a+b+c+d-1,a+\mathrm{i}x\\\
a+c,a+d\end{matrix}\,\bigg{|}\,1\bigg{)}.$ (3.2.4)

Assiotis et al. proceed by deriving a differential equation characterising the
eigenvalue density $\rho^{(Cy)}(\lambda;N,2)$ of the CyUE, using this
differential equation to obtain recurrence (3.1.13) on the sums of moments
$\mu_{k}^{(Cy)}$ (3.1.6), and then confirming that this recurrence also
characterises the continuous Hahn polynomials. Thus, they show that

$\mu_{k}^{(CyUE)}=\frac{\Gamma\left(k+1/2\right)\Gamma\left(\alpha-k-1/2\right)}{\Gamma\left(\alpha+3/2\right)\Gamma\left(-\alpha-1/2\right)\sqrt{\pi}}\frac{\mathrm{i}^{1-N}}{2}\Gamma\left(1/2-\alpha-N\right)\alpha(2\alpha+N)\\\
\times
S_{N-1}\left(-\mathrm{i}(k+1);1,\frac{1}{2}+\alpha,1,\frac{1}{2}+\alpha\right),$
(3.2.5)

with $S_{N-1}$ defined by equation (3.2.4).

We provide now the analogue of the above result of Assiotis et al. in the case
of the symmetric shifted Jacobi unitary ensemble corresponding to the weight
$w^{(sJ)}(\lambda)\big{|}_{a=b}$ (3.0.6). Let us first recall from the third
point of Remark 2.1 that $\rho^{(sJUE)}(\lambda)\big{|}_{a=b}$ is equal to
$(1-\lambda^{2})^{a}$ multiplied by a polynomial of order $N-1$. Thus, we see
from equations (2.2.8) and (3.1.5) that

$\tilde{\mu}_{k}:=\frac{\Gamma(k+N+a+3/2)}{\Gamma(k+1/2)}\mu_{k}^{(sJUE)}\Big{|}_{a=b}$
(3.2.6)

is a polynomial in $k$ of order $N-1$. Substituting this scaling into
recurrence (3.1.7) results in the simpler recurrence

$(2k+4)[(2k+3)-2(a+N)]\tilde{\mu}_{k+1}+2[(2k+2)^{2}-2N(N+2a)]\tilde{\mu}_{k}\\\
+(2k)(2(N+a+k)+1)\tilde{\mu}_{k-1}=0$ (3.2.7)

valid for $k\in\mathbb{C}$ with $\mathrm{Re}(k)>1/2$. Now, let us constrain
the continuous Hahn polynomials (3.2.4) by defining

$s_{n}(x;a,b):=S_{n}(x;a,b,\overline{a},\overline{b})$ (3.2.8)

and observe that this polynomial satisfies the recurrence [1,
18.22.13–18.22.15]

$A(x)s_{n}(x+\mathrm{i};a,b)-\left[A(x)+C(x)-n(n+2\mathrm{Re}(a+b)-1)\right]s_{n}(x;a,b)\\\
+C(x)s_{n}(x-\mathrm{i};a,b)=0,$ (3.2.9)

with

$A(x)=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),\qquad
C(x)=(x-\mathrm{i}\overline{a})(x-\mathrm{i}\overline{b}).$

######  Proposition 3.8.

For $k\in\mathbb{C}$ with ${\rm Re}(k)>-1/2$ and $\tilde{\mu}_{k}$ defined by
equation (3.2.6), we have

$\tilde{\mu}_{k}={\tilde{\mu}_{0}\mathrm{i}^{1-N}\over
N(3/2-(a+N))^{(N-1)}}s_{N-1}\Big{(}\mathrm{i}(k+1);1,{1\over 2}-(a+N)\Big{)},$
(3.2.10)

with

$\tilde{\mu}_{0}={\Gamma(N+a+3/2)\over\Gamma(1/2)}\Big{(}{2N(a+N)(2a+N)\over
1-4(a+N)^{2}}\Big{)}.$ (3.2.11)

Thus, the difference of moments ${\mu}_{k}^{(sJUE)}$ of the symmetric shifted
Jacobi unitary ensemble is given in terms of the continuous Hahn polynomials
according to

${\mu}_{k}^{(sJUE)}={\tilde{\mu}_{0}\mathrm{i}^{1-N}\Gamma(k+1/2)\over
N\Gamma(k+N+a+3/2)(3/2-(a+N))^{(N-1)}}s_{N-1}\Big{(}\mathrm{i}(k+1);1,{1\over
2}-(a+N)\Big{)}.$ (3.2.12)

###### Proof.

Equation (3.2.11) is obtained by combining equations (3.1.8) and (3.2.6).
Comparing equations (3.2.7) and (3.2.9) shows that

$q(k;a,N):=C_{N,a}\,s_{N-1}\Big{(}\mathrm{i}(k+1);1,{1\over 2}-(a+N)\Big{)},$

with $C_{N,a}$ independent of $k$, satisfies the recurrence (3.1.8). To obtain
agreement with $\tilde{\mu}_{k}$ at $k=0$, we set

$C_{N,a}={\tilde{\mu}_{0}\over s_{N-1}\Big{(}\mathrm{i};1,{1\over
2}-(a+N)\Big{)}}={\tilde{\mu}_{0}\,\mathrm{i}^{1-N}\over
N(3/2-(a+N))^{(N-1)}},$

where the second equality follows from definition (3.2.4) of the continuous
Hahn polynomials. With this choice, both $q(k;a,N)$ and $\tilde{\mu}_{k}$ are
polynomials of order $N-1$ in $k$ that coincide at $k=0$. As they both satisfy
recurrence (3.1.8), they also coincide at $k\in\mathbb{Z}$. This agreement is
extended to $k\in\mathbb{C}$ via Carlson’s theorem (recall Remark 2.5).
Finally, equation (3.2.12) follows from substituting equation (3.2.6) into
equation (3.2.10). ∎

###  3.3 The Isserlis–Wick Theorem and Ribbon Graphs

A major motivation for studying ribbon graphs (recall Definition 3.1) is their
relation to topological (hyper)maps, which we briefly discuss throughout this
section. For now, we note that the term ‘map’ is not synonymous with
‘function’ or ‘mapping’. Instead, topological maps are so-called because they
are reminiscent of the geographic maps of countries that one would find in an
atlas — they are graphs embedded in surfaces. These topological maps have been
studied by combinatorialists and geometers since the 1960s [302], [198], but
became prominent in algebraic geometry around 1984 due to Grothendieck’s work
on dessins d’enfants (children’s drawings) [284] — technically, a topological
map is equivalent to a ‘clean’ dessin (see Remark 3.6 at the end of §3.3.1).

The connection between ribbon graphs (equivalently, topological maps) and
random matrix theory is that, through consequences of the Isserlis–Wick
theorem that we will soon expound, random matrix theory provides a neat method
for enumerating ribbon graphs; the upshot here is that the pairwise polygon-
edge identifications represented by the ribbons of a ribbon graph can be
interpreted as pairings of random matrix entries. Algebraic geometers and
mathematical physicists have used this relationship in a number of intriguing
ways, of which we mention a select few:

  1. 1.

Let $\mathcal{M}_{g}^{s}$ denote the moduli space of genus $g$ compact Riemann
surfaces with $s$ marked points (i.e., the $(6g-6+2s)$-dimensional space of
parameters that determine the complex structure of such surfaces). Using the
uniqueness of the Jenkins–Strebel quadratic differential on compact Riemann
surfaces [193], [291], [292], Harer [173] showed how to canonically assign a
genus $g$, orientable, connected, metrised ribbon graph built from $s$
polygons to each point of $\mathcal{M}_{g}^{s}$ (here, metrised means that
each ribbon is labelled with a length). This had the consequence of endowing
$\mathcal{M}_{g}^{s}$ with the structure of a simplicial complex (each
$k$-simplex corresponding to a $k$-ribbon graph representative of all metrised
ribbon graphs that reduce to it upon ‘forgetting’ ribbon lengths), which
enabled Harer and Zagier [174] to reduce the problem of computing the
(orbifold) Euler characteristic of $\mathcal{M}_{g}^{s}$ to that of
enumerating ribbon graphs. This treatment was extended to the moduli space of
real algebraic curves in [165], [166], with the relevant ribbon graphs now
allowed to be non-orientable.

  2. 2.

In string theory (see, e.g., [274] for an introduction), one replaces
particles with strings, which are topologically lines or circles. The
trajectory of a string is called a worldsheet, which is a surface with
boundaries. Considering only closed (circular) strings, a worldsheet is a
surface of, say, genus $g$ with $s$ punctures — the handles counted by the
genus arise from strings splitting and combining, while the punctures are
simply contractions of the boundaries that correspond to the initial and final
states of strings. In the theory of 2D quantum gravity, one considers a
generic worldsheet with its structure, that is, a choice of metric, left
undetermined. The partition function is formulated as an integral over the
space of all possible worldsheet metrics, with the integrand given in terms of
aspects of the worldsheet that depend on the choice of metric. This integral
is difficult to make sense of because the space of worldsheet metrics is
unwieldy for many reasons. In 1990, progress was made by the independent works
[52], [89], [168], where the idea was to discretise the worldsheet and
approximate it by triangulations (i.e., topological maps with trivalent
vertices or ribbon graphs built purely from triangles). When considering the
limit of infinitely many triangles, these discrete approximations improve to
the point that each choice of triangulation corresponds to a worldsheet
metric. Consequently, the relevant partition function can essentially be
written as a sum over orientable, connected ribbon graphs.

  3. 3.

Another approach to 2D quantum gravity, called topological gravity, has
partition function equal to a generating function for certain intersection
numbers. These intersection numbers are given by integrals over the
compactifications $\overline{\mathcal{M}}_{g}^{s}$ of the moduli spaces
defined in point 1 above. A 1991 conjecture of Witten [322] is that
topological gravity should be equivalent to the formalism outlined in point 2
above. This conjecture was famously proven by Kontsevich the following year
[211] using the ideas of Harer that we reviewed in point 1. In short,
Kontsevich used the structure of $\mathcal{M}_{g}^{s}$ as a simplicial complex
(with each simplex corresponding to a specific ribbon graph) in order to
relate the aforementioned intersection numbers to explicit sums over ribbon
graphs.

For detailed surveys of these topics, see [83], [250], [220], [248].

The studies listed above share the philosophy of reducing a difficult problem
to that of enumerating ribbon graphs, and then using random matrix theory to
perform said enumerations. In contrast to this philosophy, the literature also
contains many examples of using ribbon graphs (and similar objects) to
diagrammatically encode calculations of the (mixed) moments and cumulants
defined through equations (3.0.1), (3.0.13), (3.0.14). Naturally, such
examples were first seen in the physics literature [304], [53], [287], [192],
since both random matrix theory and diagrammatic theories [179], [54] were
prevalent in the field during the mid-twentieth century. In particular, Wick’s
theorem [312] on reducing products of creation and annihilation operators to
sums of products of creation-annihilation operator pairings was well known to
quantum field theorists and easily extends to treatment of similar such
products of random matrix entries. In fact, the required generalisation
pertaining to products of normally distributed random variables was actually
known much earlier to probability theorists as Isserlis’ theorem [186]. Let us
now state a version of this theorem that is suitable for our purposes.

######  Theorem 3.1 (Isserlis–Wick).

For $k\in\mathbb{N}$, let $x_{1},\ldots,x_{k}$ be centred normal variables.
Then, if $k$ is odd, the expectation $\left<x_{1}\cdots x_{k}\right>$ is equal
to zero, while for $k$ even, it decomposes as

$\left<x_{1}\cdots
x_{k}\right>=\sum_{\sigma\in\mathfrak{P}_{k}}\left<x_{\sigma(1)}x_{\sigma(2)}\right>\cdots\left<x_{\sigma(k-1)}x_{\sigma(k)}\right>,$
(3.3.1)

where $\mathfrak{P}_{k}$ is the set of all permutations of $\\{1,\ldots,k\\}$
such that for all odd integers $i,j$ with $i<j$, $\sigma(i)<\sigma(i+1)$ and
$\sigma(i)<\sigma(j)$.

The key idea of this section, in conjunction with the Isserlis–Wick theorem,
is that for random matrices $X$ that can be expressed as sums and products of
Ginibre matrices (recall Definition 1.3), $\operatorname{Tr}\,X^{k}$ is given
by a sum of products of normally distributed random variables. Thus, recalling
Definition 1.4, the (mixed) moments and cumulants of the GOE and LOE can be
simplified using the Isserlis–Wick theorem; graphically, the covariances
$\langle x_{\sigma(j)}x_{\sigma(j+1)}\rangle$ in the summand of equation
(3.3.1) are to be represented by ribbons connecting edges labelled
$x_{\sigma(j)}$ and $x_{\sigma(j+1)}$. By linearity of the expectation, the
Isserlis–Wick theorem also holds when the variables $x_{1},\ldots,x_{k}$ are
drawn from mean-zero complex normal distributions, so it can be used to study
the GUE and LUE, as well. Similar reasoning applies in the GSE and LSE cases,
too [251], [57], though we do not discuss these cases for brevity (instead, we
appeal to the $\beta\leftrightarrow 4/\beta$ duality of Lemma 1.4).

Unfortunately, the Isserlis–Wick theorem cannot be used to relate ribbon
graphs to any of the other ensembles studied thus far in this thesis. Indeed,
the Gaussian $\beta$ ensembles with $\beta=2/3,6$ do not have matrix
realisations in terms of Ginibre matrices and must instead be understood
through the general-$\beta$ constructions of §1.2.4. Likewise, there is no
relation between the Cauchy and Ginibre ensembles, while matrix realisations
of the Jacobi ensembles involve inverse Ginibre matrices (see Definition 1.4
and Remark 1.4). However, it should be noted that the combinatorial approach
of [326] was used in [25] to express $m_{k}^{(JOE)}$ in terms of the negative-
integer moments of the LOE; the latter do not have any known ribbon graph
interpretations (see §4.4.3 for some further discussion on this point).

In §3.3.1, we show how the Isserlis–Wick theorem leads to interpretations of
the (mixed) moments and cumulants of the GUE in terms of (globally) orientable
ribbon graphs, and then explain how the GOE analogues are similarly related to
locally orientable ribbon graphs. In §3.3.2, we repeat these exercises for the
LUE and LOE, detailing a necessary bicolouring constraint on the vertices of
the polygons that are present in the relevant ribbon graphs. Finally, in
§3.3.3, we combine the ideas of §3.3.1 and §3.3.2 to derive ribbon graph
representations for the mixed moments and cumulants of the Hermitised and
antisymmetrised matrix products introduced in Section 1.3, thereby justifying
the existence of the large $N$ expansions (1.1.21) for the corresponding
connected $n$-point correlators $W_{n}$ (this latter point being significant
in the context of Chapter 4). We will be relaxed with our referencing since
our discussion will be an amalgamation of new results, results already cited,
and results that are most likely known to the community but unpublished in
full detail. That said, let us note that the exposition of the first half of
§3.3.1 is equivalent to that of [36], [329], while the first half of §3.3.2
contains statements that loosely follow from [82]. The second half of both of
these subsections discuss concepts that have been studied and reviewed in
[57], [219] and references therein. The contents of §3.3.3 are original and
based on the ongoing work [75].

####  3.3.1 Moments of the Gaussian unitary and orthogonal ensembles

Let us now demonstrate how the Isserlis–Wick theorem can be used to express
the spectral moments $m_{k}^{(GUE)}$ as sums over certain $(k/2)$-ribbon
graphs. Afterwards, we illustrate how these ideas extend to the mixed moments
and cumulants of the GUE and then show how to tweak our arguments to make them
suitable for the GOE case. At the end of this subsection, we also discuss
connections to topological and combinatorial (hyper)maps.

##### Ribbon graphs for the GUE spectral moments

Let $H$ be drawn from the $N\times N$ Gaussian unitary ensemble. According to
the convention established in Section 1.2, this means that the entries
$\\{H_{ij}\\}_{i,j=1}^{N}$ of the complex Hermitian matrix $H$ are centred
normal variables with covariances given by

$\left<H_{ij}H_{kl}\right>=\frac{1}{2}\chi_{k=j}\chi_{l=i},$ (3.3.2)

where we recall that the indicator function $\chi_{A}$ equals one when $A$ is
true and zero otherwise. Here and throughout the remainder of this subsection,
we suppress the fact that our averages are with respect to the p.d.f.
$P^{(G)}(H)$ given in equation (1.2.1). Our first order of business is to use
the formula (3.3.2) to give a multi-sum expression for the GUE spectral
moments.

######  Lemma 3.1.

Let $k\in\mathbb{N}$, recall the definition of $\mathfrak{P}_{k}$ from Theorem
3.1, and set $i_{k+1}:=i_{1}$. The spectral moment $m_{k}^{(GUE)}$ is zero if
$k$ is odd, while for even $k$,

$m_{k}^{(GUE)}=2^{-k/2}\sum_{\sigma\in\mathfrak{P}_{k}}\sum_{i_{1},\ldots,i_{k}=1}^{N}(\chi_{i_{\sigma(1)}=i_{\sigma(2)+1}}\chi_{i_{\sigma(2)}=i_{\sigma(1)+1}})\cdots(\chi_{i_{\sigma(k-1)}=i_{\sigma(k)+1}}\chi_{i_{\sigma(k)}=i_{\sigma(k-1)+1}}).$
(3.3.3)

###### Proof.

Taking equation (1.1.15) as our definition for $m_{k}^{(GUE)}$, it is well
known that

$\displaystyle m_{k}^{(GUE)}$
$\displaystyle=\left<\operatorname{Tr}\,H^{k}\right>$
$\displaystyle=\left<\sum_{i_{1},\ldots,i_{k}=1}^{N}H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots
H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}\right>$
$\displaystyle=\sum_{i_{1},\ldots,i_{k}=1}^{N}\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots
H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}\right>;$ (3.3.4)

the second line can be checked via straightforward induction, while the third
line follows from linearity of the expectation. The summand on the right-hand
side of equation (3.3.4) is the expected value of a product of normally
distributed random variables, so the Isserlis–Wick theorem applies. Hence, it
vanishes for odd $k$, and for even $k$, we have

$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots
H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}\right>\\\
=\sum_{\sigma\in\mathfrak{P}_{k}}\left<H_{i_{\sigma(1)}i_{\sigma(1)+1}}H_{i_{\sigma(2)}i_{\sigma(2)+1}}\right>\cdots\left<H_{i_{\sigma(k-1)}i_{\sigma(k-1)+1}}H_{i_{\sigma(k)}i_{\sigma(k)+1}}\right>.$
(3.3.5)

Substituting this expression into equation (3.3.4), interchanging the order of
summation, and using equation (3.3.2) then produces the desired result. ∎

The inner sum in equation (3.3.3) reduces to
$\sum_{j_{1},\ldots,j_{l}=1}^{N}1=N^{l}$ for some $1\leqslant l\leqslant
k/2+1$ since many of the indices $i_{1},\ldots,i_{k}$ must be identified to
ensure that the product of indicator functions is non-zero. To better
understand this statement, it is convenient to recast it in terms of ribbon
graphs. This can be done in a natural way by recognising that each choice of
$\sigma\in\mathfrak{P}_{k}$ determines a partitioning of the set
$\\{H_{i_{1}i_{2}},H_{i_{2}i_{3}},\ldots,H_{i_{k}i_{1}}\\}$ into $k$ disjoint
pairs: To begin, we encode the fact that the first index of each element of
$\\{H_{i_{1}i_{2}},H_{i_{2}i_{3}},\ldots,H_{i_{k}i_{1}}\\}$ is equal to the
second index of one other element of the same set by drawing a $k$-gon whose
vertices are labelled $i_{1},i_{2},\ldots,i_{k}$ in clockwise order; each edge
of the $k$-gon can be oriented in two ways, with $(i_{l}\to i_{l+1})$
representing the matrix entry $H_{i_{l}i_{l+1}}$ and $(i_{l+1}\to i_{l})$
representing $\overline{H}_{i_{l}i_{l+1}}$. Next, for each $l=1,\ldots,k/2$,
we represent the identification of the oriented polygon edges
$(i_{\sigma(2l-1)}\to i_{\sigma(2l-1)+1})$ and $(i_{\sigma(2l)+1}\to
i_{\sigma(2l)})$, implied by the factor
$\chi_{i_{\sigma(2l-1)}=i_{\sigma(2l)+1}}\chi_{i_{\sigma(2l)}=i_{\sigma(2l-1)+1}}$
in the right-hand side of equation (3.3.3), by connecting them with an
untwisted ribbon.

Figure 3.4: These are the ribbon graph representations of the terms given in
equations (3.3.7)–(3.3.9) below. The purple (grey) ribbons represent the first
(second) factor in the summands of the corresponding equations.

######  Example 3.2.

Let us illustrate the above formalism in the $k=4$ case. The set
$\mathfrak{P}_{4}$ contains three permutations: the identity, $(2,4,3)$, and
$(2,3)$. Hence, equation (3.3.3) tells us that

$m_{4}^{(GUE)}=\frac{1}{4}\left(S_{\mathrm{id}}+S_{(2,4,3)}+S_{(2,3)}\right)$
(3.3.6)

with

$\displaystyle S_{\mathrm{id}}$
$\displaystyle:=4\sum_{i_{1},\ldots,i_{4}=1}^{N}\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\right>\left<H_{i_{3}i_{4}}H_{i_{4}i_{1}}\right>=\sum_{i_{1},\ldots,i_{4}=1}^{N}(\chi_{i_{1}=i_{3}}\chi_{i_{2}=i_{2}})(\chi_{i_{3}=i_{1}}\chi_{i_{4}=i_{4}}),$
(3.3.7) $\displaystyle S_{(2,4,3)}$
$\displaystyle:=4\sum_{i_{1},\ldots,i_{4}=1}^{N}\left<H_{i_{1}i_{2}}H_{i_{4}i_{1}}\right>\left<H_{i_{2}i_{3}}H_{i_{3}i_{4}}\right>=\sum_{i_{1},\ldots,i_{4}=1}^{N}(\chi_{i_{1}=i_{1}}\chi_{i_{4}=i_{2}})(\chi_{i_{2}=i_{4}}\chi_{i_{3}=i_{3}}),$
(3.3.8) $\displaystyle S_{(2,3)}$
$\displaystyle:=4\sum_{i_{1},\ldots,i_{4}=1}^{N}\left<H_{i_{1}i_{2}}H_{i_{3}i_{4}}\right>\left<H_{i_{2}i_{3}}H_{i_{4}i_{1}}\right>=\sum_{i_{1},\ldots,i_{4}=1}^{N}(\chi_{i_{1}=i_{4}}\chi_{i_{3}=i_{2}})(\chi_{i_{2}=i_{1}}\chi_{i_{4}=i_{3}}).$
(3.3.9)

These sums are then represented by the ribbon graphs in Figure 3.4. Seen as
two-dimensional surfaces, they have, respectively, three, three, and one
boundaries. This reflects the fact that $i_{3}\equiv i_{1}$ in the first
ribbon graph, $i_{4}\equiv i_{2}$ in the second ribbon graph, and $i_{4}\equiv
i_{3}\equiv i_{2}\equiv i_{1}$ in the third ribbon graph. Hence,

$\displaystyle S_{\mathrm{id}}$
$\displaystyle=\sum_{i_{1},i_{2},i_{4}=1}^{N}1=N^{3},$ (3.3.10) $\displaystyle
S_{(2,4,3)}$ $\displaystyle=\sum_{i_{1},i_{2},i_{3}=1}^{N}1=N^{3},$ (3.3.11)
$\displaystyle S_{(2,3)}$ $\displaystyle=\sum_{i_{1}=1}^{N}1=N,$ (3.3.12)

so that by equation (3.3.6), $m_{4}^{(GUE)}=N^{3}/2+N/4$.

The computation of the above example extends quite simply to the case of $k$
being a general positive even integer: One need only interpret
$\mathfrak{P}_{k}$ within Lemma 3.1 as the set of all orientable
$(k/2)$-ribbon graphs that can be built from a single $k$-gon.

######  Lemma 3.2.

Let $k\in 2\mathbb{N}$ and $\mathfrak{P}_{k}$ be as in Theorem 3.1. Then, we
have

$m_{k}^{(GUE)}=2^{-k/2}\sum_{\sigma\in\mathrm{P}_{k}}N^{V(\sigma)},$ (3.3.13)

where $V(\sigma)$ is equal to the number of boundaries of the ribbon graph
constructed from $\sigma$ according to the prescription above Figure 3.4.

###### Proof.

Inspection of the ribbon graph corresponding to $\sigma$ shows that each
indicator function in the summand of equation (3.3.3) is represented by the
side of a ribbon identifying two vertices of the $k$-gon at the centre of the
ribbon graph. Thus, the number of unique summation indices
$i_{1},\ldots,i_{k}$ is equal to $V(\sigma)$, the number of boundaries of said
ribbon graph. Hence, we see the reduction

$\sum_{i_{1},\ldots,i_{k}=1}^{N}(\chi_{i_{\sigma(1)}=i_{\sigma(2)+1}}\chi_{i_{\sigma(2)}=i_{\sigma(1)+1}})\cdots(\chi_{i_{\sigma(k-1)}=i_{\sigma(k)+1}}\chi_{i_{\sigma(k)}=i_{\sigma(k-1)+1}})=\sum_{j_{1},\ldots,j_{V(\sigma)}=1}^{N}1=N^{V(\sigma)}.$

Substituting this into equation (3.3.3) produces the sought result. ∎

######  Remark 3.2.

Recalling that ribbons represent edge identifications of the underlying
$k$-gons, $V(\sigma)$ is also equal to the number of distinct vertices
retained by a $k$-gon after identifying edges according to the ribbon graph
corresponding to $\sigma$; see Figure 3.5 below.

Figure 3.5: These compact orientable surfaces (sphere, sphere, and torus) are
obtained by enlargening the white squares in Figure 3.4 while collapsing the
ribbons therein to their now-identified ends. Three, three, and one distinct
vertices survive the edge identification procedure, respectively.

It is evident from equation (3.3.13) that $m_{k}^{(GUE)}$ is a polynomial in
$N$. In fact, according to the statement (which we are yet to fully justify)
following the proof of Lemma 3.1, it is a polynomial of degree $k/2+1$. This
means that equation (3.3.13) can be rewritten as

$m_{k}^{(GUE)}=2^{-k/2}\sum_{V=1}^{k/2+1}N^{V}\,\\#\\{\sigma\in\mathfrak{P}_{k}\,|\,V(\sigma)=V\\},\quad
k\in 2\mathbb{N},$ (3.3.14)

where $\\#\mathcal{S}$ denotes the number of elements in the set
$\mathcal{S}$. With our current definitions of the function $V(\sigma)$,
computing the coefficient of $N^{V}$ on the right-hand side of equation
(3.3.14) requires us to generate all orientable $(k/2)$-ribbon graphs that can
be built from a $k$-gon and then count how many of them have exactly $V$
boundaries. However, we are able to give a much neater interpretation of these
polynomial coefficients, while also justifying our assumption that the degree
of $m_{k}^{(GUE)}$ in $N$ is $k/2+1$, by classifying our ribbon graphs with
respect to the genera of the corresponding compact surfaces.

Let us proceed by first observing that rather than collapsing the ribbons of a
ribbon graph (as described in Figure 3.5), another way of obtaining the
compact surface represented by a connected ribbon graph is to glue open disks
along the boundaries of said ribbon graph. This is seen to be true by noting
that contracting the disks in the resulting polygonised surface to points
while also collapsing the ribbons to edges is equivalent to just collapsing
the ribbons in the original ribbon graph (see Figure 3.6 below). Hence,
connected ribbon graphs can be drawn on the compact surfaces that they
represent and, moreover, these surfaces are of minimal (Euler) genus such that
this is possible without self-intersections (if one were able to draw a ribbon
graph on a surface of lower (Euler) genus without self-intersections, then
shrinking the ribbons and disks bounded by the ribbons in the described way
could not produce the compact surface that the ribbon graph represents). As
one would expect, this means that the (Euler) genus of a ribbon graph (recall
Definition 3.1) is equal to that of the compact surface it represents.

Figure 3.6: Gluing a pink, green, and blue disk to the boundaries of the
ribbon graph illustrated on the left produces the polygonised sphere in the
middle. Shrinking the ribbons to edges and (deformed) disks to vertices while
enlargening the white square produces the illustration on the right, which is
exactly the first surface presented in Figure 3.5.

Recall that the Euler characteristic of a compact surface is given by the
classical formula

$\chi=F-E+V,$

where $F,E,V$ are respectively the number of faces, edges, and vertices of a
chosen polygonisation of the surface. In our case, the surfaces of interest
are constructed by gluing together pairs of edges of a $k$-gon, so an obvious
choice of polygonisation for a given surface is the edge-identified $k$-gon
that it is equivalent to. Thus, we have $F=1$ and $E=k/2$ since our $k$-gons
have one face and $k$ edges which are identified pairwise into $k/2$ distinct
edges (cf. Figure 3.5). Combining this with Euler’s formula and noting that
the Euler characteristic of a compact orientable surface is also given by
$\chi=2-2g$, where $g$ is the genus of the surface, we have

$V=1+k/2-2g.$ (3.3.15)

Thus, we may reformulate equation (3.3.14) as a genus expansion.

######  Proposition 3.9.

Let $k\in\mathbb{N}$ and let $\mathfrak{P}_{k}$ be as in Theorem 3.1. If $k$
is odd, the spectral moment $m_{k}^{(GUE)}$ is zero, while if $k$ is even, we
have that

$m_{k}^{(GUE)}=2^{-k/2}\sum_{l=0}^{k/2}N^{1+k/2-l}\,\\#\\{\sigma\in\mathfrak{P}_{k}\,|\,g(\sigma)=l/2\\},$
(3.3.16)

where $g(\sigma)$ is equal to the genus of the ribbon graph labelled by
$\sigma$.

###### Proof.

Lemma 3.2 implies equation (3.3.14) with the possible requirement that the
upper terminal of the sum be changed. Equation (3.3.15) and the discussion
above Figure 3.6 tells us that the coefficient of $N^{V}$ on the right-hand
side of equation (3.3.14) is equal to the number of genus $(1+k/2-V)/2$,
connected, orientable ribbon graphs that consist of a single $k$-gon and $k/2$
ribbons. Moreover, since $g\geqslant 0$, we see that $V\leqslant k/2+1$, which
establishes that the upper limit of the sum in equation (3.3.14) is indeed
$k/2+1$; it is a simple exercise to check that the identity permutation
$\sigma=\mathrm{id}\in\mathfrak{P}_{k}$ corresponds to a (genus zero) sphere,
so that $V(\mathrm{id})=k/2+1$. Letting $l=2g$ and changing the summation
index in equation (3.3.14) according to equation (3.3.15) gives equation
(3.3.16), as desired. ∎

Replacing $k$ by $2k$ in equation (3.3.16) gives a proof of equation (3.0.8),
i.e., equation (1.2.89) of Lemma 1.3, in the GUE case and confirms our claim
following Figure 3.1 that $2^{k}M_{k,l}^{(GUE)}$ counts the number of
orientable genus $l/2$ ribbon graphs that can be constructed from a $2k$-gon.
Furthermore, since $g\in\mathbb{N}$ and $l=2g$, $M_{k,l}^{(GUE)}=0$ for $l$
odd and $m_{2k}^{(GUE)}$ is consequently an odd polynomial in $N$, in keeping
with the second point of Remark 2.1. Finally, let us point out that the
interpretation of $M_{k,l}^{(GUE)}$ in terms of ribbon graphs extends to a
representation of the generating functions $W_{1}^{(GUE),l}(x)$ (recall the
discussion at the beginning of §3.1.2) as genus $l/2$ compact orientable
surfaces with one hole — if one can replace the hole of such a representation
of $W_{1}^{(GUE),l}(x)$ by a $2k$-gon for some positive integer $k$ and
connect the edges of the $2k$-gon with ribbons drawn on the surface without
self-intersections such that excising the resulting ribbon graph yields a
disjoint union of sets homeomorphic to open disks, then that ribbon graph
contributes to the value of $M_{k,l}^{(GUE)}$.

Figure 3.7: The sphere with a hole on the left represents
$W_{1}^{(GUE),0}(x)$. On the right, we have a collection of ribbon graphs
drawn on this sphere with the hole replaced by a suitable $2k$-gon. The
illustrations on the right contribute to the value of $M_{k,0}^{(GUE)}$ with
$k=1,2,2,3$, respectively. Therefore, they also contribute to the coefficients
of the $1/x$ expansion of $W_{1}^{(GUE),0}(x)$ since the latter is essentially
a generating function for the $M_{k,0}^{(GUE)}$.

######  Remark 3.3.

  1. 1.

Recalling definition (3.1.19) of $\tilde{m}_{2k}^{(G)}$, Proposition 3.9 tells
us that

$\tilde{m}_{2k}^{(GUE)}=2^{-k}\sum_{l=0}^{k}N^{-l}\,\\#\\{\sigma\in\mathfrak{P}_{2k}\,|\,g(\sigma)=l/2\\}.$

Thus, from equation (1.1.32),
$\tilde{M}_{2k,0}^{(GUE)}=\lim_{N\to\infty}\tilde{m}_{2k}^{(GUE)}=M_{k,0}^{(GUE)}$
is equal to $2^{-k}$ multiplied by the number of genus zero (i.e., planar)
ribbon graphs that can be built from a $2k$-gon. Hence, from the discussion
contained in the caption of Figure 3.2 and the paragraph preceding it,
$2^{k}\tilde{M}_{2k,0}^{(GUE)}$ is equal to the $k\textsuperscript{th}$
Catalan number. Consequently, $W_{1}^{(GUE),0}(x)$, as described below
equation (3.1.19), is essentially a generating function for the Catalan
numbers, which is well known [290] to be given by

$W_{1}^{(GUE),0}(x)=\tfrac{1}{2}(x-\sqrt{x^{2}-2})$

after appropriate scaling (see also differential equation (2.4.17)). Of
course, applying the Sokhotski–Plemelj inversion formula (1.1.25) yields the
Wigner semi-circle law (1.2.14). This proof of the Wigner semi-circle law is a
rewording of Wigner’s original proof [313], which used the so-called method of
moments.

  2. 2.

The above proof can be extended to show that the Wigner semi-circle law holds
for the Hermitian Wigner matrix ensemble, which is represented by the same
matrices as the GUE except that the moments $\langle
H_{ij}^{p_{1}+1}H_{ji}^{p_{2}+1}\rangle$ ($p_{1},p_{2}\in\mathbb{N}$) no
longer need to be zero whenever $p_{1},p_{2}>0$. Setting rigour aside (see,
e.g., [21, Ch. 2] for a detailed treatment), the idea is that when $H$ is a
Hermitian Wigner matrix, the right-hand side of equation (3.3.5) needs to be
replaced by a sum over all partitions of $k$, rather than just pair
partitions. As an example, the contribution of the terms
$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\right>\left<H_{i_{3}i_{4}}H_{i_{4}i_{5}}H_{i_{5}i_{6}}H_{i_{6}i_{1}}\right>$
and
$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}H_{i_{3}i_{4}}\right>\left<H_{i_{4}i_{5}}H_{i_{5}i_{6}}H_{i_{6}i_{1}}\right>$
to the value of $\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots
H_{i_{5}i_{6}}H_{i_{6}i_{1}}\right>$ must now be taken into account. However,
since these terms are interpreted as products of indicator functions which
constrain our summation indices $i_{1},\ldots,i_{6}$, only pair partitions
contribute at leading order —
$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}H_{i_{3}i_{4}}H_{i_{4}i_{5}}\right>$ is
non-zero only if $i_{3},i_{4},i_{5}$ depend on $i_{1},i_{2}$ while
$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\right>\left<H_{i_{3}i_{4}}H_{i_{4}i_{5}}\right>$
being non-zero merely constrains $i_{3},i_{5}$ to depend on
$i_{1},i_{2},i_{4}$. Thus, at leading order in $N$, the spectral moments of
the Hermitian Wigner matrix $H$ are given by the Catalan numbers up to some
normalisation factor, and the argument given above follows through.

##### Ribbon graphs for the GUE mixed moments and cumulants

Letting $H$ be again drawn from the $N\times N$ GUE, the mixed moments of the
GUE (3.0.13) are given by

$m_{k_{1},\ldots,k_{n}}^{(GUE)}=\left<\prod_{i=1}^{n}\operatorname{Tr}\,H^{k_{i}}\right>,\quad
k_{1},\ldots,k_{n}\in\mathbb{N}.$

The analogue of equation (3.3.4) is then

$m_{k_{1},\ldots,k_{n}}^{(GUE)}=\sum_{i_{1}^{(1)},\ldots,i_{k_{1}}^{(1)}=1}^{N}\cdots\sum_{i_{1}^{(n)},\ldots,i_{k_{n}}^{(n)}=1}^{N}\left\langle\left(H_{i_{1}^{(1)}i_{2}^{(1)}}H_{i_{2}^{(1)}i_{3}^{(1)}}\cdots
H_{i_{k_{1}}^{(1)}i_{1}^{(1)}}\right)\cdots\right.\\\
\left.\cdots\left(H_{i_{1}^{(n)}i_{2}^{(n)}}H_{i_{2}^{(n)}i_{3}^{(n)}}\cdots
H_{i_{k_{n}}^{(n)}i_{1}^{(n)}}\right)\right\rangle.$ (3.3.17)

The summand on the right-hand side of this equation is the expected value of a
product of centred normal variables, so it can be simplified using the
Isserlis–Wick theorem, in a similar manner to equation (3.3.5). Thus,
$m_{k_{1},\ldots,k_{n}}^{(GUE)}$ vanishes if $k_{1}+\cdots+k_{n}$ is an odd
integer, whereas for $k_{1}+\cdots+k_{n}$ an even integer, we can express
$m_{k_{1},\ldots,k_{n}}^{(GUE)}$ in terms of ribbon graphs: For each
$j=1,\ldots,n$, the set
$\\{H_{i_{1}^{(j)}i_{2}^{(j)}},H_{i_{2}^{(j)}i_{3}^{(j)}},\ldots,H_{i_{k_{j}}^{(j)}i_{1}^{(j)}}\\}$
is represented by a $k_{j}$-gon with vertices labelled
$i_{1}^{(j)},i_{2}^{(j)},\ldots,i_{k_{j}}^{(j)}$ in clockwise order, while the
factors $\langle
H_{i_{p}^{(j)}i_{p+1}^{(j)}}H_{i_{q}^{(l)}i_{q+1}^{(l)}}\rangle$ (setting
$i_{k_{j}+1}^{(j)}:=i_{1}^{(j)}$) in the sum of products of covariances
produced by the Isserlis–Wick theorem are represented by untwisted ribbons
connecting the appropriate polygon edges.

Figure 3.8: In computing
$m_{4,3,3}^{(GUE)}=\left<\operatorname{Tr}(H^{4})\operatorname{Tr}(H^{3})\operatorname{Tr}(H^{3})\right>$,
one first draws a square and two triangles with vertices labelled as shown.
Then, one must list out all possible ways of pairing the polygon edges using
untwisted ribbons. The illustrated planar ribbon graph represents $\langle
H_{i_{1}^{(1)}i_{2}^{(1)}}H_{i_{2}^{(1)}i_{3}^{(1)}}\rangle\langle
H_{i_{3}^{(1)}i_{4}^{(1)}}H_{i_{4}^{(1)}i_{1}^{(1)}}\rangle\langle
H_{i_{2}^{(2)}i_{3}^{(2)}}H_{i_{3}^{(2)}i_{1}^{(2)}}\rangle\langle
H_{i_{1}^{(2)}i_{2}^{(2)}}H_{i_{3}^{(3)}i_{1}^{(3)}}\rangle\langle
H_{i_{1}^{(3)}i_{2}^{(3)}}H_{i_{2}^{(3)}i_{3}^{(3)}}\rangle$, which, when
summed over the indices, contributes a value of $N^{6}$ to
$m_{4,3,3}^{(GUE)}$.

As a slight abuse of notation, let us now define
$\mathfrak{P}_{k_{1},\ldots,k_{n}}$, for positive integers
$k_{1},\ldots,k_{n}$ such that $k_{1}+\cdots+k_{n}$ is even, to be the set of
all $\tfrac{1}{2}(k_{1}+\cdots+k_{n})$-ribbon graphs that can be constructed
by drawing $n$ polygons with respectively $k_{1},k_{2},\ldots,k_{n}$ edges and
then connecting said edges with untwisted ribbons. Then, equation (3.3.14)
extends to the form

$m_{k_{1},\ldots,k_{n}}^{(GUE)}=2^{-\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{V=1}^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})+n}N^{V}\,\\#\\{\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}\,|\,V(\Gamma)=V\\},$
(3.3.18)

where $V(\Gamma)$ is equal to the number of boundaries of $\Gamma$, which is
equivalent to the number of distinct polygon vertices retained by $\Gamma$
upon collapsing its ribbons in order to identify their ends (e.g., in Figure
3.8 above, collapsing the ribbons à la Figure 3.5 results in there being six
distinct vertices labelled by $i_{1}^{(1)}\equiv i_{3}^{(1)}$, $i_{2}^{(1)}$,
$i_{4}^{(1)}$, $i_{1}^{(2)}\equiv i_{2}^{(2)}\equiv i_{1}^{(3)}\equiv
i_{3}^{(3)}$, $i_{3}^{(2)}$ and $i_{2}^{(3)}$). When we discussed the $n=1$
case earlier, we saw that the equivalent function $V(\sigma)$ could be
expressed in terms of topological invariants of the ribbon graph corresponding
to $\sigma$. We would now like to see if such relations can be replicated in
the general-$n$ case and moreover justify the upper limit of the sum in
equation (3.3.18).

Let $\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}$ and let $\Sigma$ denote the
surface obtained by collapsing the ribbons of $\Gamma$ such that the ribbon-
ends are identified. As in the $n=1$ case, the Euler characteristic of
$\Sigma$ is given by the classical formula $\chi=F-E+V$, where $F,E,V$ are
again the number of faces, edges, and vertices of any polygonisation of
$\Sigma$. Polygonising $\Sigma$ by the polygons of $\Gamma$ with their edges
identified according to the ribbon data of $\Gamma$, we have $F=n$, the number
of polygons in $\Gamma$, and $E=(k_{1}+\cdots+k_{n})/2$, half the total number
of polygon edges in $\Gamma$. Hence, the number of vertices, $V=V(\Gamma)$, is
given by

$V(\Gamma)=\chi+\tfrac{1}{2}(k_{1}+\cdots+k_{n})-n.$ (3.3.19)

Writing $C(\Sigma)$ for the number of connected components of $\Sigma$ and
$g_{i}\geqslant 0$ for the genus of the $i\textsuperscript{th}$ such
component, we have by the additive property of the Euler characteristic that
$\chi=\sum_{i=1}^{C(\Sigma)}(2-2g_{i})$. It follows that $\chi\leqslant 2n$
since $1\leqslant C(\Sigma)\leqslant n$. Combined with the above equation,
this means that $V(\Gamma)\leqslant\tfrac{1}{2}(k_{1}+\cdots+k_{n})+n$, which
is precisely the upper limit of the sum in equation (3.3.18). However, unlike
in the $n=1$ case, this upper limit cannot always be attained. Indeed, if any
of the $k_{i}$ are odd, the corresponding polygon must be ribbon-connected to
at least one other polygon, so $\Gamma$, equivalently $\Sigma$, cannot have
$n$ connected components. This slight complication in determining the degree
of $m_{k_{1},\ldots,k_{n}}^{(GUE)}$ as a polynomial in $N$, in addition to the
reasons listed below, suggests that we should somehow focus on studying
connected ribbon graphs.

  1. 1.

If we define

$\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}:=\\{\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}\,|\,\Gamma\textrm{
is connected}\\},$

we have by the arguments above that for all
$\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$,

$V(\Gamma)\leqslant\tfrac{1}{2}(k_{1}+\cdots+k_{n})+2-n.$

Furthermore, this relation is a strict equality if $\Gamma$ is planar (genus
zero) — it is easy to convince oneself that there exists at least one such
$\Gamma$ in each $\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$.

  2. 2.

Connected surfaces have well-defined genera, so we can rewrite the analogue of
equation (3.3.18) with $\mathfrak{P}_{k_{1},\ldots,k_{n}}$ replaced by
$\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$ as a genus expansion, in a similar
fashion to equation (3.3.16).

  3. 3.

No generality is lost by studying connected ribbon graphs as every ribbon
graph is a countable disjoint union of its connected components, which is easy
to construct.

To expand on the third point given above, it is well known to
combinatorialists [290, Ch. 5] that if the mixed moment
$m_{k_{1},\ldots,k_{n}}$ on the left-hand side of the moment-cumulants
relation (3.0.14) is the weighted count of some structures of a given type
(graphs, marked surfaces, etc.) built from $n$ objects, then the corresponding
cumulants $c_{k_{1},\ldots,k_{n}}$ are the analogous counts of the subsets of
such structures that are connected. This can be understood in the context of
ribbon graphs from the following inductive argument: If we assume for all
$p<n$ and $k_{1},\ldots,k_{p}\in\mathbb{N}$ such that $k_{1}+\cdots+k_{p}$ is
an even integer that the mixed cumulants $c_{k_{1},\ldots,k_{p}}$ enumerate
(with suitable weights) the number of connected ribbon graphs that can be
built from $p$ polygons with respectively $k_{1},\ldots,k_{p}$ sides, then the
product $c_{k_{1},\ldots,k_{p}}c_{l_{1},\ldots,l_{q}}$ ($q<n$) counts the
number of ribbon graphs consisting of two connected components with the first
being a ribbon graph built from $p$ polygons with $k_{1},\ldots,k_{p}$ sides
and the second component being a ribbon graph built from $q$ polygons with
$l_{1},\ldots,l_{q}$ sides. Extending this reasoning to general products, we
see that the sum

$m_{k_{1},\ldots,k_{n}}-c_{k_{1},\ldots,k_{n}}=\sum_{\begin{subarray}{c}K\vdash\\{k_{1},\ldots,k_{n}\\}\\\
K\neq\\{\\{k_{1},\ldots,k_{n}\\}\\}\end{subarray}}\prod_{\kappa_{i}\in
K}c_{\kappa_{i}}$

counts the number of disconnected ribbon graphs that can be constructed from
$n$ polygons with $k_{1},\ldots,k_{n}$ sides. Since $m_{k_{1},\ldots,k_{n}}$
enumerates both the disconnected and connected ribbon graphs that can be built
from $n$ polygons with $k_{1},\ldots,k_{n}$ sides, it follows that
$c_{k_{1},\ldots,k_{n}}$ must enumerate all of the connected ribbon graphs
that can be constructed in said manner.

######  Remark 3.4.

If $m_{k_{1},\ldots,k_{n}}$ vanishes whenever $k_{1}+\cdots+k_{n}$ is odd, so
too does $c_{k_{1},\ldots,k_{n}}$. This can be seen from the inverse of the
moment-cumulants relation (cf. equation (1.1.30)),

$c_{k_{1},\ldots,k_{n}}=\sum_{K\vdash\\{k_{1},\ldots,k_{n}\\}}(-1)^{\\#K-1}(\\#K-1)!\prod_{\kappa_{i}\in
K}m_{\kappa_{i}}.$ (3.3.20)

Having established that we are interested in enumerating connected, orientable
ribbon graphs and that these enumerations are given by the mixed cumulants of
the GUE, let us now give the general-$n$ analogue of Proposition 3.9.

######  Proposition 3.10.

Let $n,k_{1},\ldots,k_{n}\in\mathbb{N}$ be such that $k_{1}+\cdots+k_{n}$ is
even and let $\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$ be the set of connected,
orientable $\tfrac{1}{2}(k_{1}+\cdots+k_{n})$-ribbon graphs built from $n$
polygons with respectively $k_{1},\ldots,k_{n}$ sides. The mixed cumulant
$c_{k_{1},\ldots,k_{n}}^{(GUE)}$ is given by

$c_{k_{1},\ldots,k_{n}}^{(GUE)}=\left(\frac{N}{2}\right)^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{l=0}^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})+1-n}N^{2-n-l}\,\\#\\{\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}\,|\,g(\Gamma)=l/2\\},$
(3.3.21)

where $g(\Gamma)$ is the genus of $\Gamma$.

###### Proof.

First, note that if $\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$, it is
connected with Euler characteristic $\chi=2-2g$, so equation (3.3.19) reads

$V(\Gamma)=\tfrac{1}{2}(k_{1}+\cdots+k_{n})+2-n-2g\leqslant\tfrac{1}{2}(k_{1}+\cdots+k_{n})+2-n.$
(3.3.22)

Thus, the analogue of equation (3.3.18) corresponding to connected ribbon
graphs is

$c_{k_{1},\ldots,k_{n}}^{(GUE)}=2^{-\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{V=1}^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})+2-n}N^{V}\,\\#\\{\Gamma\in\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}\,|\,V(\Gamma)=V\\}.$
(3.3.23)

Setting $l=2g$ and changing summation index according to equation (3.3.22) in
equation (3.3.23) concludes the proof. ∎

In comparing equations (3.3.18) and (3.3.21), we see that a benefit of
studying the mixed cumulants is that upon scaling them by a factor of
$N^{-(k_{1}+\cdots+k_{n})/2}$ (equivalent to considering the cumulants of the
GUE scaled by replacing the matrix $H$ in $P^{(G)}(H)$ (1.2.1) by $\sqrt{N}H$
as is consistent with Definition 1.6), they are ${\rm O}(N^{2-n})$, while the
mixed moments are ${\rm O}(N^{n})$ when scaled in this manner. Thus, the
connected $n$-point correlators $\tilde{W}_{n}^{(GUE)}$ of the global scaled
GUE, which are generating functions for the scaled cumulants, have large $N$
expansions of the form given in Theorem 1.2, whereas the corresponding
unconnected $n$-point correlators $\tilde{U}_{n}^{(GUE)}$ (1.1.26) do not. In
keeping with the discussion surrounding Figure 3.7, let us also mention that
it is convenient to represent the correlator expansion coefficients
$W_{n}^{(GUE),l}$ defined in Theorem 1.2 as genus $l/2$ compact, connected,
orientable surfaces with $n$ holes.

##### Ribbon graphs in the GOE case

The calculations pertaining to the GUE that have been outlined thus far in
this subsection follow through in much the same way when $H$ is drawn from the
$N\times N$ Gaussian orthogonal ensemble. Indeed, when $H$ is an $N\times N$
GOE matrix, the right-hand side of equation (3.3.4) now equals $m_{k}^{(GOE)}$
with the summand still simplifying to the form given in equation (3.3.5) when
$k$ is a positive even integer (and vanishing if $k$ is odd). The key
difference in studying the GOE is that equation (3.3.2) must be replaced with

$\left<H_{ij}H_{kl}\right>=\frac{1}{4}\left(\chi_{k=j}\chi_{l=i}+\chi_{k=i}\chi_{l=j}\right),$
(3.3.24)

which reflects the fact that $H$ is now a real symmetric matrix with off-
diagonal entries having variance $1/4$ (when $i\neq j$, $\langle
H_{ij}^{2}\rangle=\left<H_{ij}H_{ji}\right>=1/4$) and diagonal entries having
variance $1/2$. To ease notation, let us define

$\xi_{ij}^{kl}(t):=(1-t)\chi_{k=j}\chi_{l=i}+t\chi_{k=i}\chi_{l=j}$ (3.3.25)

so that the right-hand side of equation (3.3.24) is equal to
$\tfrac{1}{4}[\xi_{ij}^{kl}(0)+\xi_{ij}^{kl}(1)]$ (the parameter $t$ can be
thought of as keeping track of orientability; cf. the role of $\gamma$ in
[166]). Then, substituting equation (3.3.24) into equation (3.3.5) shows that
for $k$ a positive even integer,

$\left<H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots
H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}\right>=2^{-k}\sum_{\sigma\in\mathfrak{P}_{k}}\sum_{t_{1},\ldots,t_{k/2}=0}^{1}\xi_{i_{\sigma(1)}i_{\sigma(1)+1}}^{i_{\sigma(2)}i_{\sigma(2)+1}}(t_{1})\cdots\xi_{i_{\sigma(k-1)}i_{\sigma(k-1)+1}}^{i_{\sigma(k)}i_{\sigma(k)+1}}(t_{k/2}).$
(3.3.26)

As in the GUE case, we now encode the index-matching of the set
$\\{H_{i_{1}i_{2}},H_{i_{2}i_{3}},\ldots,H_{i_{k}i_{1}}\\}$ by drawing a
$k$-gon with vertices labelled $i_{1},i_{2},\ldots,i_{k}$ in clockwise order,
and then represent the edge-matching implied by the summand of equation
(3.3.26) by drawing $k/2$ ribbons connecting the edges of said $k$-gon. The
factors
$\xi_{i_{\sigma(2l-1)}i_{\sigma(2l-1)+1}}^{i_{\sigma(2l)}i_{\sigma(2l)+1}}(t_{l})$
($1\leqslant l\leqslant k/2$) in the right-hand side of equation (3.3.26) are
equivalent to the products
$\chi_{i_{\sigma(2l-1)}=i_{\sigma(2l)+1}}\chi_{i_{\sigma(2l)}=i_{\sigma(2l-1)+1}}$
seen in equation (3.3.3) when we set $t_{l}=0$. Thus, the term
$\xi_{i_{\sigma(2l-1)}i_{\sigma(2l-1)+1}}^{i_{\sigma(2l)}i_{\sigma(2l)+1}}(0)$
is represented by an untwisted ribbon connecting the oriented polygon edges
$(i_{\sigma(2l-1)}\to i_{\sigma(2l-1)+1})$ and $(i_{\sigma(2l)+1}\to
i_{\sigma(2l)})$. On the other hand, the term
$\xi_{i_{\sigma(2l-1)}i_{\sigma(2l-1)+1}}^{i_{\sigma(2l)}i_{\sigma(2l)+1}}(1)$
is non-zero if the edge $(i_{\sigma(2l-1)}\to i_{\sigma(2l-1)+1})$ is
identified to $(i_{\sigma(2l)}\to i_{\sigma(2l)+1})$, so we represent it by a
Möbius half-twisted ribbon connecting these two edges.

Figure 3.9: These ribbon graphs represent the term
$\xi_{i_{1}i_{2}}^{i_{2}i_{3}}(t_{1})\xi_{i_{3}i_{4}}^{i_{4}i_{1}}(t_{2})$ for
various choices of $t_{1},t_{2}\in\\{0,1\\}$, with the purple (grey) ribbons
representing the first (second) factor. In order, the graphs correspond to
setting $(t_{1},t_{2})=(0,1),(1,0),$ and $(1,1)$; they can be obtained by
twisting either or both of the ribbons of the left image of Figure 3.4, which
incidentally corresponds to setting $(t_{1},t_{2})=(0,0)$.

Extending the above reasoning to products of the form presented in the summand
of the right-hand side of equation (3.3.17) shows that computing the mixed
moments and cumulants of the GOE amounts to enumerating locally orientable
ribbon graphs of specific types. The necessary arguments for obtaining
explicit expressions for these moments and cumulants are the same as in the
GUE case, with the key numeric still being the number of boundaries of each
ribbon graph. Without reiterating any further details, let us simply state the
GOE analogue of Proposition 3.10.

######  Proposition 3.11.

Let $n,k_{1},\ldots,k_{n}\in\mathbb{N}$ be such that $k_{1}+\cdots+k_{n}$ is
even and let $\tilde{\mathfrak{P}}_{k_{1},\ldots,k_{n}}^{c}$ be the set of all
connected, locally orientable $\tfrac{1}{2}(k_{1}+\cdots+k_{n})$-ribbon graphs
that can be built from $n$ polygons with respectively $k_{1},\ldots,k_{n}$
edges. The mixed cumulant $c_{k_{1},\ldots,k_{n}}^{(GOE)}$ is a degree
$\tfrac{1}{2}(k_{1}+\cdots+k_{n})+2-n$ polynomial in $N$ given by the formula

$c_{k_{1},\ldots,k_{n}}^{(GOE)}=\left(\frac{N}{4}\right)^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{l=0}^{\tfrac{1}{2}(k_{1}+\cdots+k_{n})+1-n}N^{2-n-l}\,\\#\\{\Gamma\in\tilde{\mathfrak{P}}_{k_{1},\ldots,k_{n}}^{c}\,|\,\tilde{g}(\Gamma)=l\\},$
(3.3.27)

where $\tilde{g}(\Gamma)$ is the Euler genus of $\Gamma$ (recall Definition
3.1).

######  Remark 3.5.

The set $\tilde{\mathfrak{P}}_{k_{1},\ldots,k_{n}}^{c}$ defined above is in
bijection with
$\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}\times\\{0,1\\}^{(k_{1}+\cdots+k_{n})/2}$
since $\Gamma\in\tilde{\mathfrak{P}}_{k_{1},\ldots,k_{n}}^{c}$ if and only if
it is the result of twisting the ribbons of some ribbon graph drawn from
$\mathfrak{P}_{k_{1},\ldots,k_{n}}^{c}$; the tuple
$(t_{1},t_{2},\ldots,t_{(k_{1}+\cdots+k_{n})/2})\in\\{0,1\\}^{(k_{1}+\cdots+k_{n})/2}$
determines which ribbons are to be twisted.

Upon setting $n=1$, comparing equation (3.3.27) to equation (3.0.8) confirms
that $M_{k,l}^{(GOE)}$ has the combinatorial interpretation described in the
paragraph below Figure 3.1 (as an aside, observe that the ribbon graph in
Figure 3.1 is genus zero and thus contributes a value of $N^{2}$ to the
computation of $c_{1,2,3}^{(GOE)}$). Furthermore, the equality

$\\{\Gamma\in\tilde{\mathfrak{P}}_{k}^{c}\,|\,\tilde{g}(\Gamma)=0\\}=\\{\Gamma\in\mathfrak{P}_{k}^{c}\,|\,g(\Gamma)=0\\}$

implies that the moment expansion coefficients $M_{k,0}^{(GUE)}$ and
$M_{k,0}^{(GOE)}$ differ only by a factor of $2^{k/2}$, so the proof of Remark
3.3 can be used to show that the global scaled eigenvalue density of the GOE
is given by the Wigner semi-circle law (1.2.14).

A key difference between $c_{k_{1},\ldots,k_{n}}^{(GUE)}$ and
$c_{k_{1},\ldots,k_{n}}^{(GOE)}$ is that, since the (Euler) genus is a non-
negative integer, the summand of the expression (3.3.21) for
$c_{k_{1},\ldots,k_{n}}^{(GUE)}$ is identically zero for odd values of $l$,
but this is not true for the corresponding expression (3.3.27) for
$c_{k_{1},\ldots,k_{n}}^{(GOE)}$. Hence, unlike in the GUE case, the
correlator expansion coefficients $W_{n}^{(GOE),l}$ do not vanish for odd
values of $l$. Another difference between $W_{n}^{(GUE),l}$ and
$W_{n}^{(GOE),l}$ is that it does not make sense to represent the latter as a
compact surface on which to draw ribbon graphs (like in Figure 3.7) since it
is a generating function for cumulants that count both orientable and non-
orientable ribbon graphs.

##### Relations to topological and combinatorial maps and hypermaps

It is oftentimes beneficial, either for visualisation or computational
purposes, to consider alternative representations of ribbon graphs. We now
outline the connection between ribbon graphs and two such representations:
topological and combinatorial maps [303], [220], [219].

######  Definition 3.2.

A topological map is a graph embedded in a compact surface such that the edges
of the graph do not intersect and excising the graph from the surface results
in a disjoint union of open sets that are homeomorphic to open disks — these
open sets are referred to as the faces of the topological map.

######  Note 3.2.

The second condition in the above definition requires that for a given
embedding to be a valid topological map, each connected component of the graph
must live on a separate connected component of the surface. Most authors
require topological maps, hence the involved graphs, to be connected. In such
settings, it suffices to define a topological map as the embedding of a graph
in a connected compact surface of minimal (Euler) genus such that the edges of
the graph do not intersect.

There are two natural ways of transforming a ribbon graph into a topological
map. In the first formalism, one glues open disks to the boundaries of a given
ribbon graph and then contracts the polygons of the ribbon graph to points
which become the vertices of a topological map. Shrinking the polygons to
vertices also has the consequence of thinning the ribbons to edges connecting
said vertices. The disks that were initially glued to the ribbon graph become
the faces of the topological map.

Figure 3.10: As in Figure 3.6, we glue three open disks (pink, green, and
blue) to the boundaries of the ribbon graph on the left in order to embed it
into a sphere (middle image). Shrinking the white square to a vertex and the
ribbons to edges incident to said vertex produces the topological map on the
right. Our usual convention, which we break here, will be to colour the
vertices black and the faces white.

In the second formalism, one also glues disks to the boundaries of the given
ribbon graph, but then contracts these disks to vertices. The ribbons then
collapse to their ends, which become the edges of the topological map, and the
polygons of the ribbon graph become the faces of the topological map. This
construction was already exemplified in Figure 3.6. The two formalisms just
described lead to topological maps that are dual to each other in the sense
that canonically follows from the duality principle well known in standard
graph theory: one can be obtained from the other by placing a vertex at the
centre of every face, drawing an edge between each pair of new vertices
whenever the corresponding faces of the original topological map share an
edge, and then deleting the edges and vertices of the original topological
map. For example, the topological maps of Figures 3.6 and 3.10 are dual. For
the remainder of this subsection, when we refer to the topological map
corresponding to a ribbon graph, we will mean the one constructed via the
first formalism (i.e., that illustrated in Figure 3.10).

It is important to observe at this point that neither of the above
constructions are injective. For example, the first two ribbon graphs
displayed in Figure 3.4, which can be distinguished due to the labelling of
the polygon vertices (that we often leave implicit), produce the same
topological map. In the case of orientable ribbon graphs, we are able to turn
the first construction into a bijection by additionally marking a half-edge of
each of the $n$ vertices in a systematic manner, thereby forming a so-called
$n$-rooted topological map [163]. One possible convention is to mark the half-
edges corresponding to the ribbon-ends $(i_{1}^{(l)}\to i_{2}^{(l)})$ and
label them by $l$; see [219], [277] for more comprehensive reviews of
($1$-)rooted topological maps.

Figure 3.11: These rooted topological maps are constructed from the ribbon
graphs of Figure 3.4 using the first prescribed formalism. The half-edges that
are root-marked red correspond to the ribbon ends that are glued to the
polygon edges $(i_{1}\to i_{2})$.

In the case of locally orientable ribbon graphs, one also needs to keep track
of local orientation at the vertices and which edges of a given topological
map correspond to twisted ribbons. This can be done by assigning orientation
to the half-edge root-markings and labelling each edge by, say, $\pm 1$, with
$+1$ indicating that the edge represents an untwisted ribbon and $-1$ instead
referring to a Möbius half-twisted ribbon (cf. [252]). Seeing as how
introducing markings and labellings to give a combinatorial flavour to our
topological maps makes the theory more tractable, one might wonder if much
more can be gained by introducing even more labels. It turns out that it is
possible to transition to a fully combinatorial theory, removing the need for
explicitly constructing graphs or surfaces altogether. Thus, we are led to
combinatorial maps, for which there are multiple definitions.

######  Definition 3.3 (Tutte ’84).

Following [303], [161, §17.10], a combinatorial map is a quadruple
$(E_{Q},\tau_{0},\tau_{1},\tau_{2})$ where

  1. 1.

$E_{Q}$ is a finite set with the number of elements being divisible by four,

  2. 2.

$\tau_{0},\tau_{1},\tau_{2}$ are fixed-point free involutions on $E_{Q}$,

  3. 3.

$\tau_{0}\tau_{1}=\tau_{1}\tau_{0}$ and $\tau_{0}\tau_{1}$ is also fixed-point
free,

  4. 4.

the group $\left<\tau_{0},\tau_{1},\tau_{2}\right>$ generated by
$\tau_{0},\tau_{1},\tau_{2}$ is transitive, meaning that it has exactly one
orbit, that being all of $E_{Q}$.

The elements of $E_{Q}$ are called quarter-edges. The orbits of $\tau_{0}$,
$\left<\tau_{0},\tau_{1}\right>$, $\left<\tau_{0},\tau_{2}\right>$, and
$\left<\tau_{1},\tau_{2}\right>$ are respectively referred to as the half-
edges, edges, vertices, and faces of the combinatorial map.

Combinatorial maps are in one-to-one correspondence with connected, $n$-rooted
topological maps that have been edge-labelled $\pm 1$ in the aforementioned
manner (the connectedness being enforced by the fourth condition in the above
definition). This correspondence follows from the suggestive names given to
the elements of $E_{Q}$ and the various orbits highlighted in Definition 3.3:
Given a suitable topological map, one must first divide each edge both width-
and length-wise to form quarter-edges and then assign these quarter-edges
distinct labels. Then, the corresponding combinatorial map has for $E_{Q}$ the
set of said quarter-edges, while the involutions $\tau_{0},\tau_{1},\tau_{2}$
on $E_{Q}$ are such that $\tau_{0}$ interchanges two quarter-edges if and only
if they belong to the same half-edge, $\tau_{1}$ interchanges two quarter-
edges if and only if they belong to the same length-wise half-edge, and
$\tau_{2}$ interchanges two quarter-edges if and only if they are incident to
the same vertex without belonging to the same half-edge and are consecutive to
each other in the cyclic ordering around that vertex. The two possible choices
of (local) orientation at the vertices of the topological map are encoded in
the cycles of $\tau_{0}\tau_{2}$.

######  Example 3.3.

The combinatorial map corresponding to Figure 3.12 below is
$(E_{Q},\tau_{0},\tau_{1},\tau_{2})$ with

$\begin{array}[]{ll}E_{Q}=\\{1,1^{\prime},2,2^{\prime},3,3^{\prime},4,4^{\prime}\\},&\tau_{0}=(1,1^{\prime})(2,2^{\prime})(3,3^{\prime})(4,4^{\prime}),\\\
\tau_{1}=(1,2^{\prime})(1^{\prime},2)(3,4^{\prime})(3^{\prime},4),&\tau_{2}=(1,4^{\prime})(1^{\prime},2)(2^{\prime},3)(3^{\prime},4).\end{array}$

Observe that the first cycle of
$\tau_{0}\tau_{2}=(1,4,3,2)(1^{\prime},2^{\prime},3^{\prime},4^{\prime})$
describes an anticlockwise ordering of the half-edges incident to the single
vertex of the topological map, while the second cycle of $\tau_{0}\tau_{2}$
describes the opposite ordering. These cyclic orderings each prescribe a
choice of orientation for the topological map.

Figure 3.12: We divide the ribbons of a ribbon graph (left) and edges of the
corresponding topological map (right) into quarters that are labelled
$1,1^{\prime},2,2^{\prime},3,3^{\prime},4,4^{\prime}$ in clockwise order. The
half-edge marked red and oriented clockwise corresponds to the ribbon-end
$(i_{1}\to i_{2})$, while the blue plus signs signify that these edges
represent untwisted ribbons.

A combinatorial map is (globally) orientable if and only if
$\left<\tau_{0}\tau_{1},\tau_{0}\tau_{2}\right>$ has two distinct orbits
[303]. If this is the case, one may remove the redundancy of $\tau_{0}$ by
identifying quarter-edges whenever they belong to the same half-edge. In the
context of Figure 3.12, this amounts to setting $j^{\prime}\equiv j$ for
$j=1,\ldots,4$. Identifying quarter-edges according to $\tau_{0}$ induces a
choice of orientation and moreover leads to a simpler combinatorial theory on
the set of half-edges $E_{H}\simeq E_{Q}/\tau_{0}$, which has half as many
elements as that considered in Definition 3.3. This simplification is to be
expected since every compact surface has an orientable double cover [176], so
a theory allowing for non-orientable objects should intuitively require twice
as many labels as an equivalent theory focusing solely on orientable
structures.

######  Definition 3.4 (Edmonds ’60).

Following [101], [220], an oriented combinatorial map is a triple
$(E_{H},\tau_{e},\tau_{v})$ where

  1. 1.

$E_{H}$ is a finite set with an even number of elements,

  2. 2.

$\tau_{v}$ is a permutation of $E_{H}$ and $\tau_{e}$ is a fixed-point free
involution on $E_{H}$,

  3. 3.

the group $\left<\tau_{e},\tau_{v}\right>$ is transitive.

The elements of $E_{H}$ are called half-edges. The cycles of $\tau_{e}$,
$\tau_{v}$, and $\tau_{v}^{-1}\tau_{e}^{-1}$ are respectively called the
edges, vertices, and faces of the oriented combinatorial map.

This definition is preferred to Definition 3.3 in the orientable case because
the correspondence between oriented combinatorial maps and orientable,
connected, $n$-rooted topological maps is somehow more transparent. Indeed, if
one labels the half-edges of such a topological map and settles on a choice of
orientation, then the corresponding oriented combinatorial map has $E_{H}$
being the set of labelled half-edges, $\tau_{e}$ being the involution that
interchanges half-edges that belong to the same edge, and $\tau_{v}$ being a
product of disjoint cycles where each cycle describes the cyclic ordering of
the half-edges incident to a vertex that is induced by the orientation of the
topological map. Each cycle of the face permutation
$\tau_{v}^{-1}\tau_{e}^{-1}$ describes a face of the topological map by
listing out the first half of each edge that borders the face when traversing
in the direction determined by the choice of orientation.

Figure 3.13: Illustrated is an oriented planar graph with labelled half-edges.
It should be thought of as being embedded in the sphere and thus equivalent to
an oriented topological map. The corresponding oriented combinatorial map is
$(E_{H},\tau_{e},\tau_{v})$ with $E_{H}=\\{1,2,\ldots,8\\}$,
$\tau_{e}=(1,2)(3,5)(4,6)(7,8)$, and $\tau_{v}=(1,8)(2,3,4)(5,7,6)$. Note that
the cycles of $\tau_{v}$ comply with the orientation of the topological map.
Likewise, the first, second, and third cycle of the face permutation
$\tau_{v}^{-1}\tau_{e}^{-1}=(1,4,7)(2,8,5)(3,6)$ respectively list out, in
appropriate cyclic order, the blue, black, and yellow halves of the edges
bordering the triangular face, the ‘external face’, and the bigonal face.

Let us now briefly review the connection between the structures discussed
above and their hypermap analogues. A topological hypermap is simply a
connected topological map whose vertices are coloured black and white (later,
we use red) in such a way that each edge connects a black vertex to a white
one [220]. In that vein, a combinatorial hypermap is the triple of Definition
3.4 except that $\tau_{e}$ is now free to be any permutation of $E_{H}$;
$E_{H}$ is then the set of edges of the hypermap, while the cycles of
$\tau_{v}$ ($\tau_{e}$) are interpreted as black vertices (white vertices or
so-called hyperedges). Thus, an oriented combinatorial map is a combinatorial
hypermap with $\tau_{e}$ constrained to be a fixed-point free involution.
Similarly, a topological hypermap with all white vertices constrained to be
bivalent is equivalent to a connected topological map — the edges of the
hypermap are interpreted as half-edges of the topological map so that each
white vertex sits at the middle of a map-edge.

######  Remark 3.6.

In the literature surrounding Grothendieck’s dessins d’enfants [284], a
topological hypermap is referred to as a dessin d’enfant, while a topological
map — seen as a topological hypermap with bivalent white vertices — is
referred to as a clean dessin d’enfant.

####  3.3.2 Moments of the Laguerre unitary and orthogonal ensembles

Since the Laguerre unitary and orthogonal ensembles are represented by
matrices that can be expressed as products of Ginibre matrices, whose entries
are normally distributed, their (mixed) moments and cumulants can also be
studied using the Isserlis–Wick theorem. Assuming mastery over the Gaussian
case, as outlined in the previous subsection, we first show how the spectral
moments $m_{k}^{(LUE)}$ can be written as sums over certain bicoloured ribbon
graphs. Then, as in the Gaussian case, we discuss how the presented ideas
extend to the mixed cumulants of the LUE and also the LOE.

##### Ribbon graphs for the LUE spectral moments

Let $G$ be drawn from the $M\times N$ complex Ginibre ensemble and throughout
this subsection, let $\left<\,\cdot\,\right>$ denote averages with respect to
the p.d.f. $P^{(Gin)}(G)$ given in Definition 1.3. According to Definition
1.4, the $N\times N$ Wishart–Laguerre matrix $W=G^{\dagger}G$ then represents
the $(M,N)$ Laguerre unitary ensemble and the spectral moments of this
ensemble are given by (1.1.15)

$\displaystyle m_{k}^{(LUE)}$
$\displaystyle=\left<\operatorname{Tr}\,W^{k}\right>,\quad k\in\mathbb{N}$
$\displaystyle=\left<\operatorname{Tr}\,(G^{\dagger}G)^{k}\right>$
$\displaystyle=\sum_{i_{1},\ldots,i_{k}=1}^{N}\sum_{j_{1},\ldots,j_{k}=1}^{M}\left<(G^{\dagger})_{i_{1}j_{1}}G_{j_{1}i_{2}}(G^{\dagger})_{i_{2}j_{2}}G_{j_{2}i_{3}}\cdots(G^{\dagger})_{i_{k}j_{k}}G_{j_{k}i_{1}}\right>,$
(3.3.28)

in analogy with equation (3.3.4). Since the real components of the entries of
$G$ are independent, centred, normal variables such that

$\left<(G^{\dagger})_{ij}G_{kl}\right>=\chi_{i=l}\chi_{j=k},$ (3.3.29)

the Isserlis–Wick theorem then shows that

$\displaystyle\left<(G^{\dagger})_{i_{1}j_{1}}G_{j_{1}i_{2}}\cdots(G^{\dagger})_{i_{k}j_{k}}G_{j_{k}i_{1}}\right>$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{k}}\prod_{l=1}^{k}\left<(G^{\dagger})_{i_{l}j_{l}}G_{j_{\sigma(l)}i_{\sigma(l)+1}}\right>$
$\displaystyle=\sum_{\sigma\in\mathfrak{S}_{k}}\prod_{l=1}^{k}(\chi_{i_{l}=i_{\sigma(l)+1}}\chi_{j_{l}=j_{\sigma(l)}}),$
(3.3.30)

where $\mathfrak{S}_{k}$ is the set of all permutations of $\\{1,\ldots,k\\}$
and we have defined $i_{k+1}:=i_{1}$. Hence, combining equations (3.3.28) and
(3.3.30) gives the analogue of Lemma 3.1 for the LUE.

######  Lemma 3.3.

Fix $k\in\mathbb{N}$, let $\mathfrak{S}_{k}$ be the set of permutations of
$\\{1,\ldots,k\\}$ and set $i_{k+1}:=i_{1}$. We have that the spectral moments
of the LUE are given by

$m_{k}^{(LUE)}=\sum_{\sigma\in\mathfrak{S}_{k}}\sum_{i_{1},\ldots,i_{k}=1}^{N}\sum_{j_{1},\ldots,j_{k}=1}^{M}\prod_{l=1}^{k}(\chi_{i_{l}=i_{\sigma(l)+1}}\chi_{j_{l}=j_{\sigma(l)}}).$
(3.3.31)

As one might expect, the summand of the right-hand side of equation (3.3.31)
can be represented by ribbon graphs. To do so, we first draw a $2k$-gon and
label its vertices $i_{1},j_{1},i_{2},j_{2},\ldots,i_{k},j_{k}$ in clockwise
order. Next, to distinguish the two index sets, we colour the vertices
labelled by the $i_{1},\ldots,i_{k}$ black and those labelled by the
$j_{1},\ldots,j_{k}$ red. Then, for $1\leqslant l\leqslant k$, edges of the
form $(i_{l}\to j_{l})$ represent $(G^{\dagger})_{i_{l}j_{l}}$, while those of
the form $(j_{l}\to i_{l+1})$ represent $G_{j_{l}i_{l+1}}$. The factors in the
summand of the right-hand side of equation (3.3.31) are thus represented by
untwisted ribbons connecting these two types of edges. The condition that the
ribbons cannot join edges of the same type is equivalent to that of only
allowing ribbon graphs that respect the colouring of the vertices.

Figure 3.14: The illustrated ribbon graphs represent the terms given in
equations (3.3.33) and (3.3.34) below. Having alternately coloured the
vertices of the squares black and red according to the formalism described
above, we see that the vertex colourings induce colours on the boundaries of
the ribbon graph. The number of black (red) boundaries is equal to the
exponent of $N$ ($M$) in the evaluations of $S_{\mathrm{id}}^{\prime}$ and
$S_{(1,2)}^{\prime}$.

######  Example 3.4.

Let us now give the LUE analogue of Example 3.2, wherein we saw that
$m_{4}^{(GUE)}$ is equal to $(N^{3}+N^{3}+N)/4$, with each of the terms
$N^{3},N$ corresponding to a ribbon graph of Figure 3.4. In the LUE case, we
compute the second spectral moment $m_{2}^{(LUE)}$. By equation (3.3.31), it
is given by

$m_{2}^{(LUE)}=S_{\mathrm{id}}^{\prime}+S_{(1,2)}^{\prime}$ (3.3.32)

with

$\displaystyle S_{\mathrm{id}}^{\prime}$
$\displaystyle:=\sum_{i_{1},i_{2}=1}^{N}\sum_{j_{1},j_{2}=1}^{M}(\chi_{i_{1}=i_{2}}\chi_{j_{1}=j_{1}})(\chi_{i_{2}=i_{1}}\chi_{j_{2}=j_{2}})=M^{2}N,$
(3.3.33) $\displaystyle S_{(1,2)}^{\prime}$
$\displaystyle:=\sum_{i_{1},i_{2}=1}^{N}\sum_{j_{1},j_{2}=1}^{M}(\chi_{i_{1}=i_{1}}\chi_{j_{1}=j_{2}})(\chi_{i_{2}=i_{2}}\chi_{j_{2}=j_{1}})=MN^{2}.$
(3.3.34)

The evaluation $S_{\mathrm{id}}^{\prime}=M^{2}N$ follows from the fact that
the product of indicator functions in equation (3.3.33) enforces the
equivalence $i_{1}\equiv i_{2}$, but otherwise does not constrain the indices
$j_{1},j_{2}$ — similarly for $S_{(1,2)}^{\prime}$. These equivalences of
summation indices can be read off from the ribbon graphs of Figure 3.14 above.

In general, the $k\textsuperscript{th}$ spectral moment for the LUE is given
by

$m_{k}^{(LUE)}=\sum_{\sigma\in\mathfrak{S}_{k}}M^{V_{r}(\sigma)}N^{V_{b}(\sigma)},$
(3.3.35)

where $V_{r}(\sigma)$ and $V_{b}(\sigma)$ are equal to the number of red,
respectively black, boundaries of the ribbon graph corresponding to $\sigma$ —
the proof of this fact is the same as for Lemma 3.2. As with the GUE case, we
are able to reformulate equation (3.3.35) as a genus expansion, so long as the
red and black vertices of the involved ribbon graphs are placed on equal
footing.

######  Proposition 3.12.

Let $k\in\mathbb{N}$ and $\mathfrak{S}_{k}$ be as in Lemma 3.3 Setting the
Laguerre parameter $a=\hat{a}N$ with $\hat{a}={\rm O}(1)$ so that
$M=(\hat{a}+1)N$, as per the convention set out in §2.4.1 and §3.1.2, we have

$m_{k}^{(LUE)}=\sum_{l=0}^{k}N^{1+k-l}\sum_{p=1}^{k}(\hat{a}+1)^{p}\,\\#\\{\sigma\in\mathfrak{S}_{k}\,|\,g(\sigma)=l/2\textrm{
and }V_{r}(\sigma)=p\\},$ (3.3.36)

where $g(\sigma)$ is again the genus of the ribbon graph corresponding to
$\sigma$ — equivalently, $g(\sigma)$ is the minimal genus of the surface on
which this ribbon graph can be embedded into without self-intersections and it
is also the genus of the surface obtained by pairwise identifying the edges of
a $2k$-gon according to the factors
$(\chi_{i_{l}=i_{\sigma(l)+1}}\chi_{j_{l}=j_{\sigma(l)}})$ in the right-hand
side of equation (3.3.31).

###### Proof.

Observe that the construction outlined above Example 3.4 describes a bijection
between $\mathfrak{S}_{k}$ and the set of orientable $k$-ribbon graphs built
from $2k$-gons with well-defined bicoloured boundaries (i.e., each ribbon has
both a black and red side, and black (red) vertices lie on black (red)
boundaries). Thus, interpreting $\mathfrak{S}_{k}$ as this latter set,
equation (3.3.35) can be rewritten as

$m_{k}^{(LUE)}=\sum_{V_{r},V_{b}=1}^{k}M^{V_{r}}N^{V_{b}}\,\\#\\{\sigma\in\mathfrak{S}_{k}\,|\,V_{r}(\sigma)=V_{r}\textrm{
and }V_{b}(\sigma)=V_{b}\\},$ (3.3.37)

where $\\#\mathcal{S}$ has the same meaning as in equation (3.3.14).
Substituting in $M=(\hat{a}+1)N$ and using equation (3.1.15) to relate
$V(\sigma)=V_{r}(\sigma)+V_{b}(\sigma)$, the total number of boundaries of the
ribbon graph labelled by $\sigma$, to $g(\sigma)$ yields equation (3.3.36). ∎

Comparing equation (3.3.36) to equation (3.0.9) shows that $M_{k,l}^{(LUE)}$
has precisely the interpretation described below Figure 3.3. Each ribbon graph
that contributes to the computation of $m_{k}^{(LUE)}$ also contributes to the
value of $m_{2k}^{(GUE)}$ upon ‘forgetting’ the bicolouring; compare Figures
3.4 and 3.14. However, not every LUE ribbon graph arises in this way, since
some orientable ribbon graphs have no valid bicolouring. For example,
colouring any vertex of the third ribbon graph displayed in Figure 3.4 red
forces the single boundary and hence all vertices to also be coloured red,
which violates the bicolouring requirement. Nonetheless, every planar ribbon
graph has a valid bicolouring, so that the genus zero ribbon graphs pertaining
to the calculation of $m_{2k}^{(GUE)}$ are in one-to-one correspondence with
the genus zero bicoloured ribbon graphs contributing to the value of
$m_{k}^{(LUE)}$. In keeping with the discussion following Proposition 3.5,
this implies the very simple relationship
$M_{k,0}^{(LUE)}=2^{k}M_{k,0}^{(GUE)}$ when $\hat{a}=0$. In combination with
the proof of the Wigner semi-circle law given in Remark 3.3, we thus have a
combinatorial proof of the Marčenko–Pastur law (1.2.15) in the regime $a={\rm
O}(1)$.

##### Ribbon graphs for the LUE and LOE mixed cumulants

The mixed moments $m_{k_{1},\ldots,k_{n}}^{(LUE)}$ and cumulants
$c_{k_{1},\ldots,k_{n}}^{(LUE)}$ of the LUE are respectively defined through
the formulae (3.0.13), (3.0.14)

$\displaystyle m_{k_{1},\ldots,k_{n}}^{(LUE)}$
$\displaystyle=\left<\prod_{i=1}^{n}\operatorname{Tr}\,W^{k_{i}}\right>,\quad
k_{1},\ldots,k_{n}\in\mathbb{N}$ (3.3.38)
$\displaystyle=\sum_{K\vdash\\{k_{1},\ldots,k_{n}\\}}\prod_{\kappa_{i}\in
K}c_{\kappa_{i}}^{(LUE)}.$ (3.3.39)

The proofs of equations (3.3.18) and (3.3.21) can be extended to give their
LUE analogues.

######  Proposition 3.13.

Let $n,k_{1},\ldots,k_{n}\in\mathbb{N}$ and define
$\mathfrak{S}_{k_{1},\ldots,k_{n}}$ to be the set of
$(k_{1}+\cdots+k_{n})$-ribbon graphs that consist of $n$ polygons with
respectively $2k_{1},2k_{2},\ldots,2k_{n}$ edges and vertices alternately
coloured black and red, whose edges are connected by untwisted ribbons that
respect the bicolouring of the polygon vertices — let
$\mathfrak{S}_{k_{1},\ldots,k_{n}}^{c}$ be the subset of connected ribbon
graphs in $\mathfrak{S}_{k_{1},\ldots,k_{n}}$. Then, taking $a=\hat{a}N$ with
$\hat{a}={\rm O}(1)$, the mixed moments and cumulants of the LUE are given by

$\displaystyle m_{k_{1},\ldots,k_{n}}^{(LUE)}$
$\displaystyle=\sum_{V=1}^{k_{1}+\cdots+k_{n}+n}N^{V}$
$\displaystyle\qquad\times\sum_{p=1}^{k_{1}+\cdots+k_{n}}(\hat{a}+1)^{p}\,\\#\\{\Gamma\in\mathfrak{S}_{k_{1},\ldots,k_{n}}\,|\,V_{r}(\Gamma)+V_{b}(\Gamma)=V\textrm{
and }V_{r}(\Gamma)=p\\},$ (3.3.40) $\displaystyle
c_{k_{1},\ldots,k_{n}}^{(LUE)}$
$\displaystyle=\sum_{l=0}^{k_{1}+\cdots+k_{n}+1-n}N^{k_{1}+\cdots+k_{n}+2-n-l}$
$\displaystyle\qquad\times\sum_{p=1}^{k_{1}+\cdots+k_{n}+1-n}(\hat{a}+1)^{p}\,\\#\\{\Gamma\in\mathfrak{S}_{k_{1},\ldots,k_{n}}^{c}\,|\,g(\Gamma)=l/2\textrm{
and }V_{r}(\Gamma)=p\\},$ (3.3.41)

where $V_{r}(\Gamma)$ and $V_{b}(\Gamma)$ are respectively the number of red
and black boundaries of the ribbon graph $\Gamma$ and $g(\Gamma)$ is the genus
of $\Gamma$.

In contrast to what is seen in the GUE case, $m_{k_{1},\ldots,k_{n}}^{(LUE)}$
is always guaranteed to be a polynomial of degree $k_{1}+\cdots+k_{n}+n$ in
$N$ since there is at least one $\Gamma\in\mathfrak{S}_{k_{1},\ldots,k_{n}}$
such that $V_{r}(\Gamma)+V_{b}(\Gamma)$ equals this degree. This $\Gamma$,
which has $n$ connected components that are each of genus zero, exists because
the polygons present in the ribbon graphs of
$\mathfrak{S}_{k_{1},\ldots,k_{n}}$ each have an even number of sides, which
allows for polygon edges to be paired without need for any ribbons connecting
two separate polygons — the GUE analogue of
$\mathfrak{S}_{k_{1},\ldots,k_{n}}$ is the set
$\mathfrak{P}_{k_{1},\ldots,k_{n}}$, which contains no ribbon graphs with $n$
connected components if any of the $k_{i}$ are odd.

The $(M,N)$ Laguerre orthogonal ensemble is represented by the $N\times N$
Wishart–Laguerre matrix $W=G^{T}G$, where $G$ is an element of the $M\times N$
real Ginibre ensemble. Its (mixed) moments and cumulants (which are specified
by equations (3.3.38), (3.3.39) with our new definition of $W$) can be
computed in the same way as in the LUE case, except that the involved ribbon
graphs are now allowed to have Möbius half-twisted ribbons. This is because
$G$ being real means that $(G^{T})_{ij}=G_{ji}$, so equation (3.3.29) should
be rewritten as

$\left<(G^{T})_{ij}G_{kl}\right>=\left<(G^{T})_{ij}(G^{T})_{lk}\right>=\left<G_{ji}G_{kl}\right>=\frac{1}{2}\chi_{i=l}\chi_{j=k};$
(3.3.42)

recalling that our ribbon graphs are built from polygons whose edges represent
entries of either $G^{T}$ or $G$, the first average in equation (3.3.42) is
represented by an untwisted ribbon connecting these two types of edges, while
the second (third) average therein is represented by a Möbius half-twisted
ribbon connecting together two edges of the $G^{T}$ ($G$) type.

######  Proposition 3.14.

Let $n,k_{1},\ldots,k_{n}\in\mathbb{N}$ and define
$\tilde{\mathfrak{S}}_{k_{1},\ldots,k_{n}}^{c}$ to be the set of locally
orientable, connected $(k_{1}+\cdots+k_{n})$-ribbon graphs built from $n$
polygons with respectively $2k_{1},2k_{2},\ldots,2k_{n}$ edges and vertices
alternately coloured black and red, whose boundaries have well-defined colours
induced by the vertex colouring. Then, in the setting $a=\hat{a}N$ with
$\hat{a}={\rm O}(1)$, we have

$c_{k_{1},\ldots,k_{n}}^{(LOE)}=\sum_{l=0}^{k_{1}+\cdots+k_{n}+1-n}\left(\frac{N}{2}\right)^{k_{1}+\cdots+k_{n}}N^{2-n-l}\\\
\times\sum_{p=1}^{k_{1}+\cdots+k_{n}+1-n}(\hat{a}+1)^{p}\,\\#\\{\Gamma\in\tilde{\mathfrak{S}}_{k_{1},\ldots,k_{n}}^{c}\,|\,\tilde{g}(\Gamma)=l\textrm{
and }V_{r}(\Gamma)=p\\},$ (3.3.43)

where $V_{r}(\Gamma)$ is the number of red boudnaries of $\Gamma$ and
$\tilde{g}(\Gamma)$ is the Euler genus of $\Gamma$.

Note that the expression (3.3.43) is very similar to that for
$c_{k_{1},\ldots,k_{n}}^{(LUE)}$ given in equation (3.3.41), with a key
difference being that the summand of equation (3.3.43) may be non-zero for odd
values of $l$, in contrast to what is seen in equation (3.3.41). Likewise, the
mixed moments of the LOE are given by a perturbation of equation (3.3.40)
involving locally orientable ribbon graphs, though we do not display it here.

Figure 3.15: The illustrated bicoloured, locally orientable ribbon graph
represents the term
$\langle(G^{T})_{i_{1}^{(1)}j_{1}^{(1)}}(G^{T})_{i_{1}^{(2)}j_{1}^{(2)}}\rangle\langle
G_{j_{1}^{(1)}i_{2}^{(1)}}G_{j_{2}^{(2)}i_{1}^{(2)}}\rangle\langle(G^{T})_{i_{2}^{(1)}j_{2}^{(1)}}G_{j_{2}^{(1)}i_{1}^{(1)}}\rangle\langle
G_{j_{1}^{(2)}i_{2}^{(2)}}(G^{T})_{i_{2}^{(2)}j_{2}^{(2)}}\rangle$, where we
have adopted the vertex labelling convention of Figure 3.8. Interpreting the
ribbons as identifying the polygon edges, we have two faces, four edges, and
four vertices (equivalently, boundaries). Thus, by the classical formula
$\chi=F-E+V=2-\tilde{g}$, this ribbon graph is an element of
$\tilde{\mathfrak{S}}_{2,2}^{c}$ with two red boundaries and Euler genus zero.
Hence, by equation (3.3.43), it contributes a value of $(\hat{a}+1)^{2}N^{4}$
to $c_{2,2}^{(LOE)}$.

Like in the Gaussian case, the generating functions
$\tilde{W}_{n}^{(LUE)},\tilde{W}_{n}^{(LOE)}$ of the corresponding mixed
cumulants scaled according to Definition 1.6 are ${\rm O}(N^{2-n})$, so they
have large $N$ expansions of the form given in Theorem 1.2. Another similarity
between the GUE and LUE is that the correlator expansion coefficients
$W_{n}^{(LUE),l}$ (recall their definition in Theorem 1.2 and cf. the
discussion following equation (3.3.21)) can be visualised as genus $l/2$
compact, connected, orientable surfaces with $n$ holes. This represents the
fact that for given $(n,l)$, $W_{n}^{(LUE),l}$ is a generating function for
all of the ribbon graphs that can be drawn on such a surface by replacing the
holes with even-sided polygons and connecting the edges of these polygons with
ribbons such that the resulting (embedded) ribbon graph satisfies the criteria
given above Figure 3.7 (the ribbon graph does not self-intersect and excising
it from the surface results in a disjoint union of sets homeomorphic to open
disks), along with the condition that the disks bounded by the ribbons can be
coherently bicoloured black and red such that each ribbon borders both a black
and red face — in the LOE case, one needs to be careful and categorise the
relevant ribbon graphs by their orientability.

##### Relations to topological and combinatorial maps and hypermaps

As was discussed in §3.3.1, alternative representations of ribbon graphs are
interesting for various reasons. The bicoloured ribbon graphs constructed
above can be represented by the topological maps introduced in Definition 3.2
in a similar fashion to that discussed below said definition, with a few
differences: Recall that a true bijection between sets of ribbon graphs and
their topological map counterparts requires that the latter be rooted (see
Figure 3.11) and that, in the locally orientable (i.e., LOE) case, each vertex
be assigned a local orientation. However, in contrast to the GOE case, we no
longer need to keep track of which map edges correspond to twisted ribbons
since this data can be recovered by checking if the ends of a map edge
correspond to polygon edges of the same or different type (recall that the
polygon edges of an LOE ribbon graph represent entries of either $G^{T}$ or
$G$) — this is assuming that the relevant topological maps are constructed
using the formalism illustrated in Figure 3.10. (As an aside, observe that
there is a bijection between $\tilde{\mathfrak{S}}_{k_{1},\ldots,k_{n}}^{c}$
and the set $\mathfrak{P}_{2k_{1},\ldots,2k_{n}}^{c}$ of connected, orientable
GUE ribbon graphs: Alternately labelling the edges of each polygon of
$\Gamma\in\mathfrak{P}_{2k_{1},\ldots,2k_{n}}^{c}$ by $G^{T}$ and $G$, with
the ‘first’ or rooted edge of each polygon labelled $G^{T}$, and then twisting
any ribbons connecting polygon edges of the same type results in a ribbon
graph that can be uniquely bicoloured in the necessary way.) When constructing
a topological map from a bicoloured ribbon graph, the bicolouring of the
boundaries of the ribbon graph extends naturally to a face bicolouring of the
resulting topological map.

In the combinatorial setting of Definition 3.3, the necessary bicolouring
constraint is enforced by requiring that $E_{Q}$ admit a partitioning
$E_{Q}=E_{r}\cup E_{b}$ into sets $E_{r}\simeq E_{Q}/\tau_{0}$ and
$E_{b}=E_{Q}\setminus E_{r}$ of red and black quarter-edges, respectively,
such that $\tau_{0}(E_{x})=E_{Q}\setminus E_{x}$ and
$\tau_{1}(E_{x})=\tau_{2}(E_{x})=E_{x}$ for $x=r,b$. The equivalent condition
for the oriented combinatorial maps of Definition 3.4 is that
$E_{H}=E_{r}^{\prime}\cup E_{b}^{\prime}$ with $E_{b}^{\prime}=E_{H}\setminus
E_{r}^{\prime}$ and
$\tau_{e}(E_{x}^{\prime})=\tau_{v}(E_{x}^{\prime})=E_{H}\setminus
E_{x}^{\prime}$, where we again take $x=r,b$. Thus, writing
$E_{r}=\\{1^{\prime},2,3^{\prime},4\\}$ shows that the topological map of
Figure 3.12 can be bicoloured in a way that relates it to the first ribbon
graph displayed in Figure 3.14, while it can be checked that the oriented
combinatorial map of Figure 3.13 cannot be bicoloured in any valid way.

Figure 3.16: Pictured is the bicoloured $2$-rooted topological map
corresponding to the ribbon graph of Figure 3.15. It has two black (depicted
as dark grey) and red faces. Since it is embedded in the sphere, it is
orientable. Nonetheless, the notches on the blue root-markings indicate that
the local orientations of the vertices are inverse to each other.

The topological maps discussed above can be simplified in an interesting way
by transforming them into topological hypermaps (whose definition we recall
from the end of §3.3.1). In the LUE case, this is done by contracting red
faces to red vertices while identifying half-edges as follows: Traverse
clockwise around each vertex, starting at the rooted half-edge, and identify
the rooted half-edge with the second visited half-edge (retaining the root-
marking), the third visited half-edge with the fourth, and so on. This
procedure results in a topological hypermap consisting of red and black
vertices that are connected by (possibly rooted) edges. The number of faces
and red vertices of the resulting topological hypermap is respectively equal
to the number of black and red faces of the original topological map.
Moreover, the act of contracting faces to vertices does not affect the genus
of the relevant surface, so the topological hypermap obtained through the
above procedure has the same genus as the bicoloured topological map (and
ribbon graph) that it corresponds to. Thus, we may rewrite equation (3.3.41)
in terms of topological hypermaps by interpreting the coefficient

$\\#\\{\Gamma\in\mathfrak{S}_{k_{1},\ldots,k_{n}}^{c}\,|\,g(\Gamma)=l/2\textrm{
and }V_{r}(\Gamma)=p\\}$

as the number of genus $l/2$ $n$-rooted topological hypermaps with $p$ red
vertices and $n$ black vertices of valency $k_{1},\ldots,k_{n}$ (it is
straightforward to see that the above construction is invertible and thus
bijective). In terms of combinatorial hypermaps, this is equivalent to
counting the number of triples $(E_{H},\tau_{e},\tau_{v})$ such that $E_{H}$
has $k_{1}+\ldots+k_{n}$ edges, $\tau_{e}$ has $p$ cycles, $\tau_{v}$ has $n$
cycles of length $k_{1},\ldots,k_{n}$, and $\tau_{v}^{-1}\tau_{e}^{-1}$ has
$k_{1}+\cdots+k_{n}+2-n-p-l$ cycles (this ensures that the number of faces is
consistent with the genus equalling $l/2$).

Figure 3.17: The topological hypermap illustrated on the right is obtained
from the topological map on the left by contracting the red faces to vertices
while making the half-edge identifications $2\equiv 1$, $4\equiv 3$,
$2^{\prime}\equiv 1^{\prime}$, and $4^{\prime}\equiv 3^{\prime}$ — note that
we retain the blue root-markings. The corresponding combinatorial map is
$(E_{H},\tau_{e},\tau_{v})$ with $E_{H}=\\{1,3,1^{\prime},3^{\prime}\\}$,
$\tau_{e}=(1,3^{\prime},1^{\prime})(3)$, and
$\tau_{v}=(1,3)(1^{\prime},3^{\prime})$. Observe that
$\tau_{v}^{-1}\tau_{e}^{-1}=(1,3^{\prime},3)(1^{\prime})$.

The procedure described above is also valid in the LOE case, but one needs to
decorate the resulting locally orientable topological hypermaps in an
appropriate manner to ensure that the mapping is injective. To understand this
requirement, let us first recall that each half-edge of a topological map is
either of type $G^{T}$ or $G$, as discussed below equation (3.3.42) (e.g., in
the left image of Figure 3.17, the half-edges $1,3,1^{\prime},3^{\prime}$ are
of type $G^{T}$, while $2,4,2^{\prime},4^{\prime}$ are of type $G$). In the
case of LUE topological maps, traversing the boundary of a red face shows that
such a boundary is a chain of half-edges with no two consecutive half-edges
being of the same type. This is not the case when dealing with LOE topological
maps since, e.g., the boundary of the large red face in Figure 3.16 has two
rooted half-edges (automatically of type $G^{T}$) meeting each other. To
account for this subtlety, we assign $\pm 1$ ‘twist’ labellings (cf. the
discussion below Figure 3.11) to the edges of our locally orientable
topological hypermaps in such a way that interchanging the type
$G^{T}\leftrightarrow G$ of map half-edges whenever they are represented by
twisted hypermap edges (labelled $-1$) results in topological maps whose red
faces are bounded by a chain of half-edges of alternating type, as seen in the
LUE case. Note that the edges labelled $-1$ are truly twisted in the sense
that the faces glued to such edges must switch sides somewhere along them; see
Figure 3.18 below.

Figure 3.18: The left image is the result of gluing a white open disk to a
($-1$-labelled) twisted edge. To make this possible, we have folded a portion
of the disk over the grey length of the edge; the green (blue) lines are
attached to the boundary of the white disk at points that are relatively close
to each other and one should be able to imagine a continuum of similar lines
in between them. Folding the disk over the grey length of the edge results in
a peak (thin black line) and trough (dotted orange line), which are better
visualised through the depicted cross sections — the first (second) cross
section refers to the leftmost (rightmost) green and blue lines in the
illustration on the left. In this case, flattening the peak and trough (i.e.,
untwisting the edge) shows that we simply have a sphere.

The alert reader will have noticed that there are multiple ways of assigning
twists to the edges of a topological hypermap so that it correctly corresponds
to a given locally orientable, bicoloured topological map (e.g., changing the
$-1$ label in Figure 3.18 to $+1$ is inconsequential). It turns out that each
such topological map corresponds to an equivalence class of $\pm 1$-labelled
topological hypermaps, where two hypermaps are said to be equivalent if one
can be obtained from the other by using local surgery to sequentially reverse
the local orientations of some vertices while inverting the signs of all edges
connected to said vertices.

Figure 3.19: On the top left, we have an LOE ribbon graph with two twisted
ribbons and a blue root-marking on its top polygon edge. Gluing a red (dark
grey) face to the red (black) boundary, then shrinking the ribbons to edges
and the white square to a black vertex results in the topological map on the
top right; the green $\otimes$ symbol denotes a cross-cap. Contracting the red
face to a vertex while identifying half-edges in the necessary manner then
results in either of the two equivalent topological hypermaps shown in the
bottom.

######  Note 3.3.

In comparing Figures 3.18 and 3.19, one observes the well known fact that
gluing an open disk to a twisted interval (where the twist can be undone
without changing its topology) results in a sphere, while doing the same to a
loop formed by one twisted and untwisted edge (essentially a Möbius strip)
results in a real projective plane.

Let us now give a reformulation of Proposition 3.14 in terms of hypermaps.

######  Proposition 3.15.

Let $n,k_{1},\ldots,k_{n}\in\mathbb{N}$ and define
$\mathfrak{T}_{k_{1},\ldots,k_{n}}$ to be the set of orientable $n$-rooted
topological hypermaps with $k_{1}+\cdots+k_{n}$ edges, any number of red
vertices, and $n$ black vertices of valency $k_{1},\ldots,k_{n}$, with each
black vertex having one root edge incident to it; to fix convention, assign
all black vertices a clockwise orientation. Next, let
$\mathfrak{T}_{k_{1},\ldots,k_{n}}^{*}$ be the set of $\pm 1$-labelled locally
orientable topological hypermaps generated by taking elements of
$\mathfrak{T}_{k_{1},\ldots,k_{n}}$, deleting their faces, labelling each of
their edges with either $\pm 1$, and then gluing new faces to the boundaries
in a manner consistent with that shown in Figure 3.18. Finally, let
$\tilde{\mathfrak{T}}_{k_{1},\ldots,k_{n}}^{*}=\mathfrak{T}_{k_{1},\ldots,k_{n}}^{*}/\sim$,
where $\sim$ denotes the equivalence relation described immediately above
Figure 3.19. Then, we have that

$c_{k_{1},\ldots,k_{n}}^{(LOE)}=\sum_{l=0}^{k_{1}+\cdots+k_{n}+1-n}\left(\frac{N}{2}\right)^{k_{1}+\cdots+k_{n}}N^{2-n-l}\\\
\times\sum_{p=1}^{k_{1}+\ldots+k_{n}+1-n}(\hat{a}+1)^{p}\,\\#\\{\Omega\in\tilde{\mathfrak{T}}_{k_{1},\ldots,k_{n}}^{*}\,|\,\tilde{g}(\Omega)=l\textrm{
and }V_{r}(\Omega)=p\\},$ (3.3.44)

where $\tilde{g}(\Omega)$ is the Euler genus of $\Omega$ and $V_{r}(\Omega)$
is the number of red vertices of $\Omega$.

As we move on to the next subsection, let us remark that the $\pm 1$-labelled
topological hypermaps of Proposition 3.15 have combinatorial analogues that
can be defined by extending Definition 3.3 to allow for hyperedges. Without
delving into the details, for the sake of brevity, let us simply state that
the key ideas are to remove the third condition of Definition 3.3 and to
interpret the orbits of $\tau_{0}$, $\left<\tau_{0},\tau_{1}\right>$, and
$\left<\tau_{0},\tau_{2}\right>$ as edges, red (white in the context of
§3.3.1) vertices or hyperedges, and black vertices, respectively.

####  3.3.3 Moments of the Hermitised and antisymmetrised matrix products

In this subsection, we amalgamate concepts from the previous two subsections
in order to derive ribbon graph interpretations for the mixed moments (3.0.13)
and cumulants (3.0.14) of the Hermitised and antisymmetrised matrix product
ensembles introduced in Section 1.3. We first give some basic results for
general products of (possibly) rectangular matrices before highlighting some
interesting simplifications in terms of topological hypermaps for the simplest
non-trivial products (i.e., the Hermitised and antisymmetrised Laguerre
ensembles corresponding to taking $m=1$ and $\nu_{1}=\nu_{0}=0$ in Definitions
1.10 and 1.11).

##### Ribbon graphs for the mixed moments of the Hermitised matrix product
ensembles

Following Definition 1.10, let $m,N_{0}\in\mathbb{N}$, fix
$N_{1},\ldots,N_{m}\in\mathbb{N}$ such that for $1\leqslant i\leqslant m$,
$N_{i}\geqslant N_{0}$, and write $N:=N_{m}$. Then, the
$(N_{0},\ldots,N_{m-1},N)$ Hermitised matrix product ensemble is represented
by the $N\times N$ product

$\mathcal{H}_{m}=G_{m}^{\dagger}\cdots G_{1}^{\dagger}HG_{1}\cdots G_{m},$
(3.3.45)

where $H$ is drawn from the $N_{0}\times N_{0}$ GUE and for $1\leqslant
i\leqslant m$, each $G_{i}$ is drawn independently from the $N_{i-1}\times
N_{i}$ complex Ginibre ensemble. Letting $\left<\,\cdot\,\right>$ now, and for
the remainder of this subsection, denote averages with respect to the j.p.d.f.

$P^{(G)}(H)\Big{|}_{N\mapsto
N_{0}}\,\prod_{i=1}^{m}P^{(Gin)}(G_{i})\Big{|}_{(M,N)\mapsto(N_{i-1},N_{i})},$

we give a preliminary characterisation of the mixed moments (3.0.13) of the
$(N_{0},\ldots,N_{m-1},N)$ Hermitised matrix product ensemble.

######  Lemma 3.4.

Fix $m,n,k_{1},\ldots,k_{n}\in\mathbb{N}$ and let $\mathcal{H}_{m}$ be as in
equation (3.3.45). For $1\leqslant s\leqslant n$ and $1\leqslant t\leqslant
k_{s}-1$, set $i_{k_{s}}^{(s;-m-1)}:=i_{1}^{(s;m+1)}$ and
$i_{t}^{(s;-m-1)}:=i_{t+1}^{(s;m+1)}$. Then, we have that

$\displaystyle m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$
$\displaystyle=\left(\prod_{a=1}^{n}\prod_{b=1}^{k_{a}}\prod_{c=1}^{m+1}\sum_{i_{b}^{(a;\pm
c)}=1}^{N_{|c-1|}}\right)\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}\right>$
$\displaystyle\hskip
120.00018pt\times\prod_{u=1}^{m}\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}(G^{\dagger}_{u})_{i_{t}^{(s;u+1)}i_{t}^{(s;u)}}(G_{u})_{i_{t}^{(s;-u)}i_{t}^{(s;-u-1)}}\right>.$
(3.3.46)

###### Proof.

Taking equation (3.0.13) as the definition of
$m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$ and inserting the expansion

$\operatorname{Tr}\,\mathcal{H}_{m}^{k_{s}}=\sum_{i_{1}^{(s)},\ldots,i_{k_{s}}^{(s)}=1}^{N}(\mathcal{H}_{m})_{i_{1}^{(s)}i_{2}^{(s)}}(\mathcal{H}_{m})_{i_{2}^{(s)}i_{3}^{(s)}}\cdots(\mathcal{H}_{m})_{i_{k_{s}}^{(s)}i_{1}^{(s)}}$

shows that

$m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}=\left<\prod_{s=1}^{n}\operatorname{Tr}\,\mathcal{H}_{m}^{k_{s}}\right>=\left(\prod_{a=1}^{n}\prod_{b=1}^{k_{a}}\sum_{i_{b}^{(a)}=1}^{N}\right)\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}(\mathcal{H}_{m})_{i_{t}^{(s)}i_{t+1}^{(s)}}\right>,$

where we have set $i_{k_{s}+1}^{(s)}:=i_{1}^{(s)}$ for all $1\leqslant
s\leqslant n$ — cf. equation (3.3.17). Making the replacement
$(\mathcal{H}_{m})_{i_{t}^{(s)}i_{t+1}^{(s)}}\mapsto(\mathcal{H}_{m})_{i_{t}^{(s;m+1)}i_{t+1}^{(s;m+1)}}=(\mathcal{H}_{m})_{i_{t}^{(s;m+1)}i_{t}^{(s;-m-1)}}$
and using the definition (3.3.45) of $\mathcal{H}_{m}$ to write

$(\mathcal{H}_{m})_{i_{t}^{(s;m+1)}i_{t}^{(s;-m-1)}}=(G_{m}^{\dagger})_{i_{t}^{(s;m+1)}i_{t}^{(s;m)}}(G_{m-1}^{\dagger})_{i_{t}^{(s;m)}i_{t}^{(s;m-1)}}\cdots\\\
\cdots(G_{1}^{\dagger})_{i_{t}^{(s;2)}i_{t}^{(s;1)}}H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}(G_{1})_{i_{t}^{(s;-1)}i_{t}^{(s;-2)}}\cdots(G_{m})_{i_{t}^{(s;-m)}i_{t}^{(s;-m-1)}}$

then shows that

$\displaystyle m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$
$\displaystyle=\left(\prod_{a=1}^{n}\prod_{b=1}^{k_{a}}\prod_{c=1}^{m+1}\sum_{i_{b}^{(a;\pm
c)}=1}^{N_{|c-1|}}\right)\left\langle\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}(G_{m}^{\dagger})_{i_{t}^{(s;m+1)}i_{t}^{(s;m)}}(G_{m-1}^{\dagger})_{i_{t}^{(s;m)}i_{t}^{(s;m-1)}}\cdots\right.$
$\displaystyle\hskip
110.00017pt\left.\cdots(G_{1}^{\dagger})_{i_{t}^{(s;2)}i_{t}^{(s;1)}}H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}(G_{1})_{i_{t}^{(s;-1)}i_{t}^{(s;-2)}}\cdots(G_{m})_{i_{t}^{(s;-m)}i_{t}^{(s;-m-1)}}\right\rangle.$

As the random matrices $H,G_{1},\ldots,G_{m}$ within equation (3.3.45) are
assumed to be (mutually) independent, the average in the above can be split
into the product of averages shown on the right-hand side of equation
(3.3.46), thereby proving the latter formula. ∎

As shown in §3.3.1 and §3.3.2, each average on the right-hand side of equation
(3.3.46) can be simplified using the Isserlis–Wick theorem. In particular, the
average

$\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}\right>$

is identically zero if it contains an odd number of entries, so
$m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$ vanishes if $k_{1}+\cdots+k_{n}$
is an odd integer. On the other hand, when $k_{1}+\cdots+k_{n}$ is an even
integer, we are led to the following ribbon graph construction: For each
$s=1,\ldots,n$, draw a $k_{s}(2m+1)$-gon and label its vertices in clockwise
order in such a way that the first $2m+1$ vertices are labelled
$i_{1}^{(s;m+1)},i_{1}^{(s;m)},\ldots,i_{1}^{(s;1)},i_{1}^{(s;-1)},i_{1}^{(s;-2)},\ldots,i_{1}^{(s;-m)}$,
the next $2m+1$ vertices are labelled
$i_{2}^{(s;m+1)},\ldots,i_{2}^{(s;1)},i_{2}^{(s;-1)},\ldots,i_{2}^{(s;-m)}$,
and so on, with the last $2m+1$ vertices being labelled
$i_{k_{s}}^{(s;m+1)},\ldots,i_{k_{s}}^{(s;1)},i_{k_{s}}^{(s;-1)},\ldots,i_{k_{s}}^{(s;-m)}$
— for $1\leqslant t\leqslant k_{s}$ and $1\leqslant u\leqslant m$, edges of
the form $(i_{t}^{(s;1)}\to i_{t}^{(s;-1)})$ represent
$H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}$, while $(i_{t}^{(s;u+1)}\to i_{t}^{(s;u)})$
and $(i_{t}^{(s;-u)}\to i_{t}^{(s;-u-1)})$ represent
$(G^{\dagger}_{u})_{i_{t}^{(s;u+1)}i_{t}^{(s;u)}}$ and
$(G_{u})_{i_{t}^{(s;-u)}i_{t}^{(s;-u-1)}}$, respectively.

Applying the Isserlis–Wick theorem to the averages on the right-hand side of
equation (3.3.46) results in a product of covariances that are exactly of the
type seen in the GUE and LUE cases and can thus be represented by untwisted
ribbons connecting polygon edges representing entries from a matrix and its
adjoint; that is, for $1\leqslant u\leqslant m$, if one end of a ribbon is
connected to an edge representing an entry of $G^{\dagger}_{u}$ ($H$), then
its other end must be connected to an edge representing an entry of $G_{u}$
($H$). This latter constraint can be satisfied by assigning, for each
$u=1,\ldots,m+1$, a unique colour $\phi_{u}$ to all polygon vertices labelled
by indices of the form $i_{t}^{(s;\pm u)}$, with $1\leqslant s\leqslant n$ and
$1\leqslant t\leqslant k_{s}$, and then only allowing ribbon graphs whose
boundaries have well-defined colours inherited from the vertices they pass
through; see Figure 3.20 below.

Figure 3.20: In computing
$m_{2,1,1}^{(\mathcal{H}_{2})}=\langle\operatorname{Tr}(\mathcal{H}_{2}^{2})\operatorname{Tr}(\mathcal{H}_{2})^{2}\rangle$,
one first draws a decagon, representing
$\operatorname{Tr}(\mathcal{H}_{2}^{2})$, and two pentagons, each representing
a factor of $\operatorname{Tr}(\mathcal{H}_{2})$, with their vertices coloured
as shown (the colours $\phi_{1},\phi_{2},\phi_{3}$ are respectively red, blue,
and green) — we do not display the vertex labellings, but mention that the top
green vertex of each polygon, moving left to right, is respectively labelled
$i_{1}^{(1;3)},i_{1}^{(2;3)},i_{1}^{(3;3)}$, while the other vertex labels
follow from the prescription given above. In the next step, one simply lists
out all possible ways of pairwise identifying the polygon edges using
untwisted ribbons whose sides inherit well-defined colours from the vertices;
the displayed ribbon graph illustrates one such pairing. For visual clarity,
and without any mathematical meaning, we have assigned the colours light blue,
purple, and grey to the ribbons with respectively two red sides, a red and
blue side, and a blue and green side. These ribbons represent pairings of the
form $\langle H_{ij}H_{ji}\rangle$,
$\langle(G_{1}^{\dagger})_{ij}(G_{1})_{ji}\rangle$, and
$\langle(G_{2}^{\dagger})_{ij}(G_{2})_{ji}\rangle$ (for suitable $i,j$),
respectively.

Extending arguments given in the previous two subsections (in particular, the
ideas underlying Examples 3.2 and 3.4) leads to an elegant formula for the
mixed moments (3.0.13) and cumulants (3.0.14) of the
$(N_{0},\ldots,N_{m-1},N)$ Hermitised matrix product ensemble in terms of
ribbon graphs, in analogy with equation (3.3.35).

######  Lemma 3.5.

Fix $m,n,k_{1},\ldots,k_{n}\in\mathbb{N}$ and let $\mathcal{H}_{m}$ be as in
equation (3.3.45). If $k_{1}+\cdots+k_{n}$ is odd, we have that

$m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}=c_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}=0.$

Thus, constraining ourselves to the case that $k_{1}+\cdots+k_{n}$ is even,
define $\mathfrak{P}^{(m)}_{k_{1},\ldots,k_{n}}$ to be the set of
$(m+1)$-coloured orientable $\tfrac{1}{2}(2m+1)(k_{1}+\cdots+k_{n})$-ribbon
graphs that can be constructed through the procedure described above Figure
3.20 and let
$\mathfrak{P}^{(m),c}_{k_{1},\ldots,k_{n}}\subseteq\mathfrak{P}^{(m)}_{k_{1},\ldots,k_{n}}$
be the subset of connected such ribbon graphs. Then,

$\displaystyle m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$
$\displaystyle=2^{-\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{\Gamma\in\mathfrak{P}^{(m)}_{k_{1},\ldots,k_{n}}}N_{0}^{V_{0}(\Gamma)}N_{1}^{V_{1}(\Gamma)}\cdots
N_{m}^{V_{m}(\Gamma)},$ (3.3.47) $\displaystyle
c_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}$
$\displaystyle=2^{-\tfrac{1}{2}(k_{1}+\cdots+k_{n})}\sum_{\Gamma\in\mathfrak{P}^{(m),c}_{k_{1},\ldots,k_{n}}}N_{0}^{V_{0}(\Gamma)}N_{1}^{V_{1}(\Gamma)}\cdots
N_{m}^{V_{m}(\Gamma)},$ (3.3.48)

where, for $0\leqslant u\leqslant m$, $V_{u}(\Gamma)$ is the number of
boundaries of $\Gamma$ that are of the colour $\phi_{u+1}$.

###### Proof.

As mentioned above, applying the Isserlis–Wick theorem to the averages in the
right-hand side of equation (3.3.46) shows that

$\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}H_{i_{t}^{(s;1)}i_{t}^{(s;-1)}}\right>$

vanishes if $k_{1}+\cdots+k_{n}$ is odd and is otherwise given by a sum over
$\mathfrak{P}_{k_{1},\ldots,k_{n}}$ (defined above equation (3.3.18)) of a
product of indicator functions identifying the summation indices
$i_{b}^{(a;\pm 1)}$. Likewise, the averages

$\left<\prod_{s=1}^{n}\prod_{t=1}^{k_{s}}(G^{\dagger}_{u})_{i_{t}^{(s;u+1)}i_{t}^{(s;u)}}(G_{u})_{i_{t}^{(s;-u)}i_{t}^{(s;-u-1)}}\right>$

are given by a similar sum, but over $\mathfrak{S}_{k_{1},\ldots,k_{n}}$
(recall Proposition 3.13). Thus,
$m_{k_{1},\ldots,k_{n}}^{(\mathcal{H}_{m})}=0$ when $k_{1}+\cdots+k_{n}$ is
odd and, otherwise, the canonical bijection
$\mathfrak{P}^{(m)}_{k_{1},\ldots,k_{n}}\simeq\mathfrak{P}_{k_{1},\ldots,k_{n}}\times(\mathfrak{S}_{k_{1},\ldots,k_{n}})^{m}$
allows us to write the product of averages in the right-hand side of equation
(3.3.46) as a sum over $\mathfrak{P}^{(m)}_{k_{1},\ldots,k_{n}}$ of a product
of indicator functions.