characteristic polynomials outside of the bulk. This bound is less explicit
than Proposition 8.1, but it is stronger and much easier to obtain.

######  Proposition 7.11.

For $\tau\in[0,1)$, $\ell>0$, and $|x|>1+\tau$, there is $C>0$ such that

$C^{-1}\leq\frac{\mathbb{E}[|\det(A_{N}-xI)|^{\ell}]}{\mathbb{E}[|\det(A_{N}-xI)|]^{\ell}}\leq
C.$ (7.61)

In addition, for fixed $k$, there is $C$ such that

$C^{-1}\leq\frac{\mathbb{E}[|\det(A_{N-k}-xI)|^{\ell}]}{\mathbb{E}[|\det(A_{N}-xI)|]^{\ell}}\leq
C.$ (7.62)

Moreover, $C$ may be chosen to be uniform in compact subsets of $x$.

###### Proof.

To show (7.61) it is sufficient to show that for $0<\epsilon<1+|\tau|-x$ there
is $C$ such that

$C^{-1}\leq\mathbb{E}[|\det(A_{N}-xI)|^{\ell}]\exp\left(-N\ell\int_{0}^{\infty}\log_{\epsilon}(y)\mathbb{E}[\nu_{A_{N},x}](dy)\right)\leq
C.$ (7.63)

The upper bound follows from Lemma 7.5. We also have by Jensen’s inequality
that

$\exp\left(N\ell\int_{0}^{\infty}\log(y)\mathbb{E}[\nu_{A_{N},x}](dy)\right)\leq\mathbb{E}[|\det(A_{N}-xI)|^{\ell}].$
(7.64)

Thus to show (7.63) it is sufficient to show that

$N|\int_{0}^{\infty}\log_{\epsilon}(y)\mathbb{E}[\nu_{A_{N},x}](dy)-\int_{0}^{\infty}\log(y)\mathbb{E}[\nu_{A_{N},x}](dy)|=$
(7.65) $N\int_{0}^{\epsilon}\log(\epsilon/y)\mathbb{E}[\nu_{A_{N},x}](dy)\to
0.$ (7.66)

We observe by the bound $s_{1}(A_{N}-xI)>s_{1}(A_{N})-x$ and Lemma 7.6 there
is $C,c>0$ such that $\mathbb{P}(\nu_{A_{N},x}([0,\epsilon])\neq 0)\leq
Ce^{-cN}$. Denoting this event by $\mathcal{A}$, we see that

$N\int_{0}^{\epsilon}\log(\epsilon/y)\mathbb{E}[\nu_{A_{N},x}](dy)=N\int_{0}^{\epsilon}\log(\epsilon/y)\mathbb{E}[I_{\mathcal{A}}\nu_{A_{N},x}](dy)\leq\mathbb{E}[I_{\mathcal{A}}\log(\min(\epsilon,|s_{N}(A_{N})|))]\leq$
(7.67)
$\mathbb{P}(\mathcal{A})^{1/2}\mathbb{E}[\log(\min(\epsilon,|s_{N}(A_{N})|))^{2}]^{1/2}.$
(7.68)

On the other hand by Lemma 7.10 we have that

$\mathbb{E}[\log(|s_{N}(A_{N})|)^{2}]=-2\int_{0}^{\epsilon}\log(s)\mathbb{P}(|s_{N}(A_{N})|<s)ds\leq
2N^{2}\int_{0}^{1}s\log(s)ds.$ (7.69)

As the right is finite, this completes the proof of (7.66).

For (7.62), we observe by (7.66) that it is sufficient to show that

$|N\int_{0}^{\infty}\log_{\epsilon}(y)\mathbb{E}[\nu_{A_{N},x}](dy)-(N-k)\int_{0}^{\infty}\log_{\epsilon}(y)\mathbb{E}[\nu_{A_{N-k},x}](dy)|\leq
C.$ (7.70)

Employing the tails estimates of Lemma 7.6 as before, we see that for $K>0$
sufficiently large

$|N\int_{K}^{\infty}\log_{\epsilon}(y)\mathbb{E}[\nu_{A_{N},x}](dy)|\to 0.$
(7.71)

As $\log_{\epsilon}$ is bounded on $[0,K]$ it suffices to show the following
total variation bound

$\sup_{a\geq
0}|N\mathbb{E}[\nu_{A_{N},x}]([0,a])-(N-k)\mathbb{E}[\nu_{A_{N-k},x}]([0,a])|\leq
C.$ (7.72)

Let us denote the principal $(N-k)$-by-$(N-k)$ submatrix of $A_{N}$ by
$\bar{A}_{N}$. We note by the interlacing property for the singular values of
a submatrix, we have for any $a\in\mathbb{R}$

$|N\nu_{A_{N},x}([0,a])-(N-k)\nu_{\bar{A}_{N},x}([0,a])|\leq k.$ (7.73)

Now noting the distributional equality

$\bar{A}_{N}\stackrel{{\scriptstyle d}}{{=}}\sqrt{\frac{N-k}{N}}A_{N-k},$
(7.74)

we see that

$\sup_{a\geq
0}|\mathbb{E}[\nu_{\bar{A}_{N},x}]([0,a])-\mathbb{E}[\nu_{A_{N-k},x}]([0,a])|\leq\left(1-\sqrt{\frac{N-k}{N}}\right)\mathbb{E}[s_{1}(A_{N-k})].$
(7.75)

Combining these observations and Lemma 7.6 completes the proof. ∎

We end this section by noting the following result which follows immediately
from the proof of Lemma 4.4 of [53].

######  Lemma 7.12.

[53] For any $|\tau|<1$, $0<\gamma<1$, and $u\in\mathbb{R}$, as well as any
choice of $u_{N}\to u$, there is $C,c>0$ such that

$\mathbb{P}\left(s_{N-i}(A_{N}-u_{N}I)>cN^{\gamma-1}\text{ for all
}N^{\gamma}\leq i<N\right)\leq Ce^{-cN^{\gamma}}.$ (7.76)

##  8 Integral Moments of the Characteristic Polynomials of Elliptic Ensembles

The main result of this section is the proof of Proposition 8.1, which gives a
bound for the integral moments of the absolute characteristic polynomial of
the Elliptic ensemble in the bulk up to a constant factor.

######  Proposition 8.1.

For $\tau\in(0,1)$, $\ell\in\mathbb{N}$, and $|x|<1+\tau$ there is $C$ such
that

$C^{-1}\exp(\ell N\phi_{\tau}(x))\leq\mathbb{E}[|\det(A_{N}-xI)|^{\ell}]\leq
CN^{\ell(\ell-1)/4}\exp(\ell N\phi_{\tau}(x)).$ (8.1)

Moreover, these bounds are uniform in compact subsets of $(-1-\tau,1+\tau)$.

Our approach will rely on a relation between the integral moments of the
characteristic polynomials and the correlation functions for the ensemble.
Such relations have proven to be a crucial tool in the study of the Hermitian
and symmetric ensembles, leading back to the fundamental works of Brézin,
Fyodorov, Hikami, Mehta, Normand, and Strahov [16, 52, 36] (see as well [45,
43]). These relations rely crucially on the invariant nature of the ensemble,
and so rely deeply on the Gaussian nature of our matrices. This reduces the
problem to obtaining an asymptotic expansion for the real correlations
functions near the diagonal, using their representations as Pfaffian processes
from [29, 28, 58].

Explicit asymptotics for the first moment of the absolute characteristic
polynomial were obtained in [32], in the course of computing the annealed
complexity. Another early result for the second moment was [3], which gave an
explicit combinatorial expression for the expectation of the characteristic
polynomials taken at two conjugate points in the complex plane. Asymptotics of
these, and related, expressions were given in [37] by applying Laplace’s
method to a clever integral representation, generalizing earlier work for the
Ginibre ensemble [33]. As for the characteristic polynomials on the real line,
and especially for moments higher than two, these expressions become
significantly more complex.

For specific non-Hermitian ensembles, particularly the Ginibre ensemble, the
ensemble has additional symmetries which allow one to generalize some of the
rich combinatorial structure from the Hermitian case. For some works in this
direction see [34, 60, 30].

We will note fix, for the remainder of the section, a choice of $1>\tau>0$ and
omit this choice from the notation for $A_{N}$. We note that as the
(algebraic) eigenvalues of $A_{N}$ solve the equation $\det(A_{N}-\lambda)=0$,
the non real eigenvalues must come in conjugate pairs. In particular, if there
are $k$ real eigenvalues of $A_{N}$, say $\lambda_{1},\dots\lambda_{k}$, then
$k$ must have the same parity as $N$ and there must be $(N-k)/2$ eigenvalues
$z_{1},\dots z_{k}\in\mathbb{H}$ such that the eigenvalues of $A_{N}$ are
given by $\lambda_{1},\dots,\lambda_{k},z_{1},\dots
z_{k},\bar{z}_{1},\dots\bar{z}_{k}$.

When $A_{N}$ is sampled from $GEE(N,\tau)$ (with $-1<\tau<1$), this $k$ is
random, and it is convenient to treat each possible value of $k$ separately.
To be more explicit, let us denote the event that $A_{N}$ has exactly $k$ real
eigenvalues as $\mathcal{A}_{N,k}$ (where $k$ is such that $N-k$ is even). On
$\mathcal{A}_{N,k}$, let us denote the real eigenvalues of $A_{N}$ as
$\\{\lambda_{i}\\}_{i=1}^{k}$ and the complex eigenvalues values of $A_{N}$ as
$\\{x_{i}+iy_{i},x_{i}-iy_{i}\\}_{i=1}^{(N-k)/2}$ with $y_{i}>0$. The density
of this process on $\mathbb{R}^{k}\times\mathbb{H}^{(N-k)/2}$ was computed in
[48] (see also (2.7) of [29]). In particular, they show that if
$f:\mathbb{R}^{k}\times\mathbb{H}^{(N-k)/2}\to\mathbb{R}$ is a function, which
is symmetric in $\mathbb{R}^{k}$ and $\mathbb{H}^{(N-k)/2}$, then

$\mathbb{E}[f(\lambda_{1},\dots\lambda_{k},x_{1}+iy_{1},\dots,x_{(N-k)/2}+iy_{(N-k)/2})I_{\mathcal{A}_{N,k}}]=$
(8.2)
$\int_{\mathbb{R}^{k}\times\mathbb{H}^{(N-k)/2}}P_{N,k}(\\{\lambda_{i}\\}_{i=1}^{k},\\{x_{i}+iy_{i}\\}_{i=1}^{k})\prod_{i=1}^{(N-k)/2}d\lambda_{i}\prod_{i=1}^{(N-k)/2}dx_{i}dy_{i},$
(8.3)

where here $P_{N,k}$ denotes the (symmetrized) probability density function

$P_{N,k}(\\{\lambda_{i}\\}_{i=1}^{k},\\{x_{i}+iy_{i}\\}_{i=1}^{(N-k)/2}):=\frac{1}{K_{N}(\tau)}|\Delta(\\{\lambda_{i}\\}_{i=1}^{k}\cup\\{x_{i}\pm
iy_{i}\\}_{i=1}^{k})|\times$ (8.4)
$\frac{1}{k!((N-k)/2)!}\prod_{i=1}^{k}e^{-\frac{N}{2(1+\tau)}\sum_{i=1}^{k}\lambda_{i}^{2}}\prod_{i=1}^{(N-k)/2}2e^{\frac{N}{(1+\tau)}(y_{i}^{2}-x_{i}^{2})}\mathrm{erfc}\left(y_{i}\sqrt{\frac{2N}{1-\tau^{2}}}\right),$
(8.5)

where $K_{N}(\tau)$ is a normalization constant given by

$K_{N}(\tau):=N^{-N(N+1)/4}(1+\tau)^{N/2}2^{N(N+1)/4}\prod_{\ell=1}^{N}\Gamma(\ell/2),$
(8.6)

and $\Delta$ denotes the Vandermonde determinant

$\Delta(x_{1},\dots,x_{k})=\prod_{i<j}(x_{i}-x_{j}).$ (8.7)

For $\ell<N$ we may define the (real) correlation function
$\rho^{\ell}_{N}:\mathbb{R}^{\ell}\to\mathbb{R}$ by

$\rho^{\ell}_{N}(w_{1},\dots
w_{\ell}):=\frac{1}{K_{N}(\tau)}\sum_{L+2M=N,\;L\geq\ell,M\geq
0}\frac{L!}{(L-\ell)!}\times$ (8.8)
$\int_{\mathbb{R}^{L-\ell}\times\mathbb{H}^{M}}P_{N,L}(\\{w_{i}\\}_{i=1}^{\ell}\cup\\{\lambda_{i}\\}_{i=1}^{L-\ell},\\{x_{i}+iy_{i}\\}_{i=1}^{M})\prod_{i=1}^{L-\ell}d\lambda_{i}\prod_{i=1}^{M}dx_{i}dy_{i}.$
(8.9)

We note that while the variable $M=(N-L)/2$ here is redundant, its inclusion
makes the expressions below significantly more manageable.

That this coincides with the alternative definition of a correlation function
(such as (5.1) of [15] or (4.2) of [29]) follows from Corollary 2 of [15].

We now demonstrate a relationship between the integral powers of the absolute
characteristic polynomial and the values of the correlation function around
the diagonal.

######  Lemma 8.2.

For any $\ell\geq 1$, $N>\ell$, and $\mu_{1}>\dots>\mu_{\ell}$, we have that

$\mathbb{E}\left[\prod_{i=1}^{\ell}|\det(A_{N}-\mu_{i}I)|\right]=D_{N,\ell}(\tau)\frac{e^{\frac{N}{2(1+\tau)}\sum_{i=1}^{\ell}\mu_{i}^{2}}}{|\Delta(\mu_{1},\dots\mu_{\ell})|}\rho_{N+\ell}^{\ell}(\tilde{\mu}_{1},\dots,\tilde{\mu}_{\ell}),$
(8.10)

where $\tilde{\mu}_{i}:=\sqrt{N/(N+\ell)}\mu_{i}$ and here

$D_{N,\ell}(\tau):=\frac{K_{N+\ell}(\tau)}{K_{N}(\tau)}\left(\frac{N+\ell}{N}\right)^{N/2+(N+\ell)(N+\ell-1)/4}.$
(8.11)

###### Proof.

We note that for any $u_{1},\dots u_{i},v_{1},\dots v_{j}\in\mathbb{C}$ we
have that

$|\Delta(u_{1},\dots,u_{i},v_{1},\dots,v_{j})|=\prod_{k=1}^{i}\prod_{l=1}^{j}|(x_{k}-y_{l})||\Delta(u_{1},\dots,u_{i})\Delta(v_{1},\dots,v_{j})|.$
(8.12)

Applying this relation to (8.9) we obtain that

$\mathbb{E}\left[\prod_{i=1}^{\ell}|\det(A_{N}-\mu_{i})|\right]e^{-\frac{N}{2(1+\tau)}\sum_{i=1}^{\ell}\mu_{i}^{2}}|\Delta(\mu_{1},\dots\mu_{\ell})|=$
(8.13) $\sum_{L,M\geq
0,L+2M=N}\frac{1}{L!M!}\frac{1}{K_{N}(\tau)}\int_{\mathbb{R}^{L}\times\mathbb{H}^{M}}|\Delta(\\{\mu_{i}\\}_{i=1}^{\ell}\cup\\{\lambda_{i}\\}_{i=1}^{L}\cup\\{x_{i}\pm
iy_{i}\\}_{i=1}^{M})|\times$ (8.14)
$\prod_{i=1}^{\ell}e^{-\frac{N}{2(1+\tau)}\mu_{i}^{2}}\prod_{i=1}^{L}e^{-\frac{N}{2(1+\tau)}\sum_{i=1}^{L}\lambda_{i}^{2}}\prod_{i=1}^{M}2e^{-\frac{N}{(1+\tau)}(y_{i}^{2}-x_{i}^{2})}\mathrm{erfc}\left(y_{i}\sqrt{\frac{2N}{1-\tau^{2}}}\right)\prod_{i=1}^{L}d\lambda_{i}\prod_{i=1}^{M}dx_{i}dy_{i}.$
(8.15)

Rescaling all of the variables by a factor of $\sqrt{N/(N+\ell)}$, we see that
the right hand side of we may write the right hand side of (8.15) as

$D_{N,\ell}(\tau)\sum_{L,M\geq 0,L+2M=N}\frac{(L+\ell)!}{L!}\times$ (8.16)
$\int_{\mathbb{R}^{L}\times\mathbb{H}^{M}}P_{N+\ell,L+\ell,M}(\\{\tilde{\mu}_{i}\\}_{i=1}^{\ell}\cup\\{\lambda_{i}\\}_{i=1}^{L},\\{x_{i}+iy_{i}\\}_{i=1}^{M})\prod_{i=1}^{L}d\lambda_{i}\prod_{i=1}^{M}dx_{i}dy_{i}.$
(8.17)

Reindexing we may finally rewrite this as

$D_{N,\ell}(\tau)\rho^{\ell}_{N+\ell}(\tilde{\mu}_{1},\dots\tilde{\mu}_{k}).$
(8.18)

∎

Given Lemma 8.2, it is clear that we are reduced to understanding
$\rho^{\ell}_{N}$, and so to progress further we will need to recall a
convenient formula for $\rho^{\ell}_{N}$. Similarly to the symmetric case,
these correlation functions can be given in terms of a certain Pfaffian
kernel. In particular, we will see below that

$\rho^{\ell}_{N}(x_{1},\dots,x_{\ell})=N^{\ell/2}\mathrm{Pf}([K_{N}(x_{i},x_{j})]_{i,j}),$
(8.19)

where $\mathrm{Pf}$ denotes the Pfaffian and $K_{N}(x,y):\mathbb{R}^{2}\to
M_{2}(\mathbb{R})$ is a certain matrix kernel satisfying
$K_{N}(x,y)=-K_{N}(y,x)^{T}$, which we will define below.

To state the main technical results of this section we will first require its
asymptotic form. We write

$\mathcal{K}(x,y):=\begin{bmatrix}-\mathcal{I}(x,y)&\mathcal{S}(x,y)\\\
-\mathcal{S}(y,x)&\mathcal{D}(x,y)\end{bmatrix},$ (8.20)

where

$\mathcal{S}(x,y):=\frac{1}{\sqrt{\pi(1-\tau^{2})}}\exp\left(-\frac{1}{2(1-\tau^{2})}(x-y)^{2}\right),$
(8.21)
$\mathcal{D}(x,y):=\partial_{x}\mathcal{D}(x,y),\;\;\mathcal{I}(x,y):=\frac{1}{2}\mathrm{Sign}(y-x)-\int_{x}^{y}\mathcal{S}(x,z)dz.$
(8.22)

The next result establishes that for $x,y\in(-1-\tau,1+\tau)$, the kernel
$\mathcal{K}(\sqrt{N}x,\sqrt{y})$ is exponentially close to $K_{N}(x,y)$.

######  Lemma 8.3.

If we denote

$R_{N}(x,y):=K_{N}(x,y)-\mathcal{K}(\sqrt{N}x,\sqrt{N}y),$ (8.23)

then for any $x,y\in(-1-\tau,1+\tau)$ there is $C,c>0$ such that

$\|R_{N}(x,y)\|_{F}\leq Ce^{-Nc}.$ (8.24)

Moreover, for any $\epsilon>0$ and $k\geq 0$ we may choose $c$ small enough
that for $x,y\in(-1-\tau+\epsilon,1+\tau-\epsilon)$ we have that

$\sum_{i,j\leq k}\|\partial_{x}^{i}\partial_{y}^{j}R_{N}(x,y)\|_{F}\leq
Ce^{-cN}.$ (8.25)

Before discussing the proof of Lemma 8.3, we will show how it may be combined
with Lemma 8.2 prove Proposition 8.1.

###### Proof of Proposition 8.1.

By 8.2 we have that

$\mathbb{E}\left[|\det(A_{N-k}-\sqrt{(N+k)/N}\mu
I)|^{k}\right]=e^{k\frac{N}{2(1+\tau)}\mu^{2}}D_{N-k,k}(\tau)\left(\lim_{\mu_{1},\dots,\mu_{k}\to\mu}\frac{\rho_{N}^{k,0}(\mu_{1},\dots,\mu_{k})}{|\Delta(\mu_{1},\dots,\mu_{k})|}\right).$
(8.26)

We will begin by understanding the last term on the right-hand side using
(8.19) and Lemma 8.3. Let us define

$\mathcal{C}_{k}(\mu_{1},\dots,\mu_{k})=\mathrm{Pf}([\mathcal{K}(\mu_{i},\mu_{j})]_{1\leq
i,j\leq k}).$ (8.27)

We will also find it convenient to define

$\bar{\mathcal{K}}_{ij}(x,y):=\begin{bmatrix}-\mathcal{I}(x,y)+\frac{1}{2}\mathrm{Sign}(y-x)-\frac{1}{2}\mathrm{Sign}(i-j)&\mathcal{S}(x,y)\\\
-\mathcal{S}(y,x)&\mathcal{D}(x,y)\end{bmatrix},$ (8.28)
$\bar{\mathcal{C}}_{k}(\mu_{1},\dots,\mu_{k})=\mathrm{Pf}([\bar{\mathcal{K}}_{ij}(\mu_{i},\mu_{j})]_{1\leq
i,j\leq k}).$ (8.29)

These functions are now analytic as functions of
$\mu_{1},\dots,\mu_{k}\in\mathbb{C}$, and we observe that for
$\mu_{1}>\dots>\mu_{k}$, we have that

$\mathcal{C}_{k}(\mu_{1},\dots,\mu_{k})=\bar{\mathcal{C}}_{k}(\mu_{1},\dots,\mu_{k}).$
(8.30)

Now as $\bar{\mathcal{C}}_{k}(\mu_{1},\dots\mu_{k})$ vanishes when
$\mu_{i}-\mu_{j}=0$ for any $i\neq j$, the function is divisible by
$\Delta(\mu_{1},\dots,\mu_{k})$, so we see that there is an analytic function
on $\mathbb{R}$, $\mathcal{R}_{k}(\mu)$, such that

$\mathcal{R}_{k}(\mu)=\lim_{\mu_{1},\dots\mu_{k}\to\mu}\frac{\mathcal{C}_{k}(\mu_{1},\dots,\mu_{k})}{\Delta(\mu_{1},\dots,\mu_{k})}$
(8.31)

where the limit here is restricted to points such that
$\mu_{1}>\dots>\mu_{k}$. Modifying $K_{N}(x,y)$ as in (8.28) we may define an
augmented kernel $\bar{K}_{N,i,j}(x,y)$, and further define

$\bar{\rho}^{k}_{N}(\mu_{1},\dots,\mu_{k})=N^{k/2}\mathrm{Pf}([\bar{K}_{N,i,j}(\mu_{i},\mu_{j})]_{1\leq
i,j\leq k}).$ (8.32)

In view of (8.19) we see that when $\mu_{1}>\dots>\mu_{k}$,

$\bar{\rho}^{k}_{N}(\mu_{1},\dots,\mu_{k})=\rho^{k}_{N}(\mu_{1},\dots,\mu_{k}).$
(8.33)

Moreover, the same argument as for $\bar{\mathcal{C}}_{k}$ shows that there is
an analytic function $R_{N,k}(\mu)$ such that

$R_{N,k}(\mu)=\lim_{\mu_{1},\dots\mu_{k}\to\mu}\frac{\rho^{k}_{N}(\mu_{1},\dots,\mu_{k})}{\Delta(\mu_{1},\dots,\mu_{k})},$
(8.34)

where again the limit is restricted as above. We will now show that for small
enough $c>0$

$|R_{N,k}(\mu)-N^{k(k+1)/4}\mathcal{R}_{k}(\sqrt{N}\mu)|=O(e^{-Nc})$ (8.35)

uniformly in compact subsets of $\mu\in(-1-\tau,1+\tau)$. To do this we
observe that if we take $\mu_{k}=\mu$ and apply Taylor’s theorem to
$\bar{\rho}^{k}_{N}(\mu_{1},\dots,\mu_{k})$ around the point
$(\mu,\dots,\mu)$, then $R_{N,k}(\mu)$ is proportional to the coefficient of
the term $\Delta(\mu_{1},\dots,\mu_{k-1},\mu)$ in the Taylor expansion. In
particular, we see that

$R_{N,k}(\mu)=\sum_{i_{1},\dots,i_{k(k-1)/2}=1}^{k}a_{i_{1},\dots,i_{k(k-1)/2}}\frac{\partial^{k(k-1)/2}}{\partial\mu_{i_{1}}\dots\partial\mu_{i_{k(k-1)/2}}}\bar{\rho}^{k}_{N}(\mu,\dots,\mu),$
(8.36)

where $a_{i_{1},\dots,i_{k(k-1)/2}}$ are some fixed universal constants.
Similarly we conclude that

$\mathcal{R}_{k}(\mu)=\sum_{i_{1},\dots,i_{k(k-1)/2}=1}^{k}a_{i_{1},\dots,i_{k(k-1)/2}}\frac{\partial^{k(k-1)/2}}{\partial\mu_{i_{1}}\dots\partial\mu_{i_{k(k-1)/2}}}\bar{\mathcal{C}}_{k}(\mu,\dots,\mu)$
(8.37)

where $a_{i_{1},\dots,i_{k(k-1)/2}}$ are the same constants.

We observe by Lemma 8.3 that for $|x|<1+\tau$ and any $n,m\geq 0$ there is
$C,c>0$ such that

$\|\frac{\partial^{n+m}}{\partial^{n}x\partial^{m}y}\left(\bar{K}_{N,i,j}(x,y)-\bar{\mathcal{K}}_{ij}(\sqrt{N}x,\sqrt{N}y)\right)\|_{F}\big{|}_{x=y}\leq
Ce^{-Nc}.$ (8.38)

Additionally, we see that

$\|\frac{\partial^{n+m}}{\partial^{n}x\partial^{m}y}\bar{K}_{N,i,j}(x,y)\|_{F}\big{|}_{x=y}\leq\|\frac{\partial^{n+m}}{\partial^{n}x\partial^{m}y}\bar{\mathcal{K}}_{ij}(\sqrt{N}x,\sqrt{N}y)\|_{F}\big{|}_{x=y}+Ce^{-Nc}\leq$
(8.39) $C(N^{n+m}+e^{-Nc}).$ (8.40)

Expanding the Pfaffian as a polynomial in its entries and applying (8.38) and
(8.40) we may write the right-hand side of (8.36) as

$\sum_{i_{1},\dots,i_{k(k-1)/2}=1}^{k}a_{i_{1},\dots,i_{k(k-1)/2}}N^{k/2}\frac{\partial^{k(k-1)/2}}{\partial\mu_{i_{1}}\dots\partial\mu_{i_{k(k-1)/2}}}\bar{C}_{N}(\sqrt{N}\mu,\dots,\sqrt{N}\mu)+O(e^{-Nc/2})=$
(8.41) $N^{k(k+1)/4}\mathcal{R}_{k}(\mu)+O(e^{-Nc/2}).$ (8.42)

This completes the proof of (8.35).

We now note by (8.37) that the function $\mathcal{R}_{k}$ is bounded. Thus we
see that for fixed $k$ there is $C>0$ such that

$\lim_{\mu_{1},\dots,\mu_{k}\to\mu}\frac{\rho^{k}_{N}(\mu_{1},\dots,\mu_{k})}{|\Delta(\mu_{1},\dots,\mu_{k})|}\leq
CN^{k(k+1)/4}.$ (8.43)

Thus by (8.26) we see that for any $\epsilon>0$ there is $C$ such that for
$N,\ell\geq 1$ and any choice of $|\mu|<1+\tau-\epsilon$

$\mathbb{E}\left[|\det(A_{N-k}-\mu I)|^{k}\right]\leq
Ce^{k\frac{N}{2(1+\tau)}\mu^{2}}D_{N-k,k}N^{k(k+1)/4}.$ (8.44)

To complete the proof, we need to understand the asymptotics for $D_{N-k,k}$.
It will be convenient to reindex our choice and instead consider the
asymptotics for $D_{N,\ell}$ for fixed $\ell\in\mathbb{Z}$. We note that

$\left(\frac{N+\ell}{N}\right)^{N/2+(N+\ell)(N+\ell-1)/4}=e^{\ell N/4}O(1).$
(8.45)

We also note that

$\frac{K_{N}(\tau)}{K_{N-1}(\tau)}=(1+\tau)^{1/2}N^{-N/2}2^{N/2}\Gamma(N/2)\left(\frac{N-1}{N}\right)^{N(N-1)/4}=$
(8.46) $O(1)e^{-N/2}N^{-1/2}e^{-N/4}=O(1)N^{-1/2}e^{-3N/4}.$ (8.47)

Thus inductively we may conclude that.

$D_{N,\ell}(\tau)=O(1)N^{-\ell/2}e^{-\ell\frac{N}{2}}.$ (8.48)

In particular, we see that

$\mathbb{E}\left[|\det(A_{N-k}-\mu I)|^{k}\right]\leq
Ce^{k\frac{N}{2(1+\tau)}\mu^{2}-\frac{N}{2}}D_{N-k,k}N^{k(k-1)/4}.$ (8.49)

Noting that for $|x|<1+\tau$,
$\phi_{\tau}(x)=\frac{x^{2}}{2(1+\tau)}-\frac{1}{2}$ completes the proof of
the upper bound of the proposition.

For the lower bound note that as

$\mathbb{E}\left[|\det(A_{N-k}-\mu
I)|\right]^{k}\leq\mathbb{E}\left[|\det(A_{N-k}-\mu I)|^{k}\right],$ (8.50)

it suffices to consider the case of $k=1$. Proceeding as above we see that
there is $c>0$ such that

$\mathbb{E}\left[|\det(A_{N}-\mu I)|\right]\geq
ce^{N\phi_{\tau}(x)}\mathcal{R}_{1}(\sqrt{N+1}\mu).$ (8.51)

On the other hand for any $x\in\mathbb{R}$

$\mathcal{R}_{1}(x)=\frac{1}{\sqrt{\pi(1-\tau^{2})}},$ (8.52)

which completes the proof. ∎

The proof of Lemma 8.3 will comprise the remainder of this section. To
proceed, we will of course have to give an explicit form for the kernel
$K_{N}(x,y)$. As is typical for the $\beta=1$ case in random matrix theory,
the definition in the odd and even case will need to be given separately. For
clarity, we will complete the proof of Lemma 8.3 in the case of even $N$
first, and then proceed to the case where $N$ is odd.

In particular, we will now assume for the moment that $N$ is even. In this
case the definition of $K_{N}$ is given in Section 4 of [29], which we now
recall. Here and below $H_{k}$ will denote the $k$-th Hermite polynomial. We
have that

$K_{N}(x,y):=\begin{bmatrix}-I_{N}(x,y)&S_{N}(x,y)\\\
-S_{N}(y,x)&D_{N}(x,y)\end{bmatrix},$ (8.53)

where

$D_{N}(x,y):=\partial_{x}S_{N}(x,y),\;\;I(x,y):=\frac{1}{2}\mathrm{Sign}(y-x)-\int_{x}^{y}S_{N}(x,z)dz,$
(8.54)
$S_{N}(x,y):=\frac{e^{-N\frac{(x^{2}+y^{2})}{2(1+\tau)}}}{\sqrt{2\pi}}\sum_{k=0}^{N-2}\frac{1}{k!}C_{k}(\sqrt{N}x)C_{k}(\sqrt{N}y)+\frac{e^{-N\frac{y^{2}}{2(1+\tau)}}}{2\sqrt{2\pi}(1+\tau)}\frac{C_{N-1}(\sqrt{N}y)\Phi_{N-2}(\sqrt{N}x)}{(N-2)!},$
(8.55)

where

$C_{k}(x):=\left(\frac{\tau}{2}\right)^{k/2}H_{k}\left(\frac{x}{\sqrt{2\tau}}\right).$
(8.56)
$\Phi_{N}(x):=\left(\sqrt{2\pi(1+\tau)}\frac{N!}{2^{N/2}(N/2)!}\right)-2\int_{x}^{\infty}e^{-t^{2}/2(1+\tau)}C_{N}(t)dt.$
(8.57)

Up to rescaling these follow from (4.4), (4.5), (4.6), (4.11), (5.16), and
(5.24). With these definitions, (8.19) occurs as their (4.3). To avoid future
confusion, we note that (8.57) only holds when $N$ is even.

We will need the following global bound on $H_{k}$.

######  Lemma 8.4.

There is $C$ such that for $x\in\mathbb{R}$

$|H_{k}(\sqrt{2k}x)|\leq C(k!)^{1/2}2^{k/2}e^{ks(x)},$ (8.58)

where here

$s(x):=x^{2}-I(|x|\geq
1)\left(|x|\sqrt{x^{2}-1}-\log(\sqrt{x^{2}-1}+|x|)\right)=$ (8.59)
$\frac{1}{2}+\log(2)+\frac{2}{\pi}\int_{-1}^{1}\sqrt{1-y^{2}}\log(|x-y|)dy.$
(8.60)

###### Proof.

Cramér’s Inequality (see [65]) states that for any $x\in\mathbb{R}$ and $k\geq
0$

$|H_{k}(\sqrt{2k}x)|\leq(k!)^{1/2}2^{k/2}e^{kx^{2}}.$ (8.61)

As $s(x)-x^{2}=0$ for $|x|<1$ and $s(x)-x^{2}=O(||x|-1|)$ elsewhere, this
gives the desired claim for $|x|<1+k^{-1}$.

For the remaining region we may employ the asymptotic expansions in Theorem 1
and Corollary 2 of [25] to find $C$ such that for $|x|>1$

$|H_{k}(\sqrt{2k}x)|\leq
Cn^{k/2}e^{-k/2}2^{k/2}e^{ks(x)}\sqrt{\left(1-|x|/\sqrt{x^{2}-1}\right)}$
(8.62)

When $|x|\geq 1+k^{-1}$, the last term is order $k^{1/4}$, so employing (8.62)
and Stirling’s approximation completes the proof. ∎

###### Proof of Lemma 8.3 with $N$ even.

Our first step is to employ Mehler’s formula (see [1]) which shows that for
$|\rho|<1$,

$\sum_{k=0}^{\infty}\frac{\rho^{k}}{2^{k}k!}H_{k}(x)H_{k}(y)=\frac{1}{\sqrt{1-\rho^{2}}}\exp\left(-\frac{\rho^{2}(x^{2}+y^{2})-2\rho
xy}{(1-\rho^{2})}\right).$ (8.63)

In particular we observe that

$\frac{e^{-\frac{1}{2(1+\tau)}(x^{2}+y^{2})}}{\sqrt{2\pi}}\sum_{k=0}^{\infty}\frac{1}{k!}C_{k}(x)C_{k}(y)=\mathcal{S}(x,y).$
(8.64)

Thus we may write

$S_{N}(x,y)-\mathcal{S}(\sqrt{N}x,\sqrt{N}y)=-\frac{e^{-\frac{N}{2(1+\tau)}(x^{2}+y^{2})}}{\sqrt{2\pi}}\sum_{k=N-1}^{\infty}\frac{1}{k!}C_{k}(\sqrt{N}x)C_{k}(\sqrt{N}y)+$
(8.65)
$\frac{e^{-N\frac{y^{2}}{2(1+\tau)}}}{2\sqrt{2\pi}(1+\tau)}\frac{C_{N-1}(\sqrt{N}y)\Phi_{N-2}(\sqrt{N}x)}{(N-2)!}.$
(8.66)

Now we see immediately from Lemma 8.4 that if we define

$h_{\tau}(x):=-\frac{x^{2}}{2(1+\tau)}+\frac{1}{2}\log(\tau)+s(\frac{x}{2\tau}),$
(8.67)

then there is $C$ such that for $x\in\mathbb{R}$

$e^{-\frac{k}{2(1+\tau)x^{2}}}|C_{k}(\sqrt{k}x)|\leq
C(k!)^{1/2}e^{kh_{\tau}(x)}.$ (8.68)

It is easily verified that $h_{\tau}(x)$ is maximized precisely at the points
$\\{\pm(1+\tau)\\}$, where $h_{\tau}(\pm(1+\tau))=0$, and that for $|x|\neq
1+\tau$ we have that $h_{\tau}(x)<0$. From this we see that for $\epsilon>0$
we may choose $c,C>0$ such that if both $|x-(1+\tau)|>\epsilon$ and
$|x+(1+\tau)|>\epsilon$

$e^{-\frac{k}{2(1+\tau)x^{2}}}|C_{k}(\sqrt{k}x)|\leq C(k!)^{1/2}e^{-ck}.$
(8.69)

For the remainder of the proof we will fix a choice of $\epsilon>0$, and
consider $x,y\in(-1-\tau+\epsilon,1+\tau-\epsilon)$ unless otherwise stated.
We also allow the constants $C,c>0$ (which are dependent on $\epsilon$) to
change from line to line. From (8.69) we see that

$|e^{-\frac{N}{2(1+\tau)}(x^{2}+y^{2})}\sum_{k=N-1}^{\infty}\frac{1}{k!}C_{k}(\sqrt{N}x)C_{k}(\sqrt{N}y)|\leq
C\sum_{k=N-1}^{\infty}e^{-2kc}\leq Ce^{-Nc}.$ (8.70)

We now focus our attention on the second term in (8.66). We note that
employing (8.68),

$|\int_{x}^{\infty}e^{-t^{2}/2(1+\tau)}C_{N}(t)dt|\leq C(N!)^{1/2}.$ (8.71)

Further employing Stirling’s formula to the constant term in (8.57) we see
that

$|\Phi_{N}(x)|\leq C(N!)^{1/2}.$ (8.72)

Combining these we conclude that

$|\frac{C_{N-1}(\sqrt{N}y)\Phi_{N-2}(\sqrt{N}x)}{(N-2)!}|\leq Ce^{-Nc}.$
(8.73)

Altogether this establishes that

$|S_{N}(x,y)-\mathcal{S}_{N}(x,y)|\leq Ce^{-Nc}.$ (8.74)

Now for $\ell\geq 0$, we may repeatedly employ the recurrence relation

$H^{\prime}_{k}(x)=2xH_{k}(x)-H_{k+1}(x),$ (8.75)

to obtain from (8.69) that there is $C>0$ such that for $x\in\mathbb{R}$

$|\frac{d^{\ell}}{dx^{\ell}}\left(e^{-\frac{k}{2(1+\tau)x^{2}}}C_{k}(\sqrt{k}x)\right)|\leq
Ck^{\ell}(k!)^{1/2}e^{kh_{\tau}(x)}.$ (8.76)

Repeating the above analysis we see that we have for any $n,m\geq 0$

$|\frac{d^{n}}{dx^{n}}\frac{d^{m}}{dy^{m}}\left(S_{N}(x,y)-\mathcal{S}(\sqrt{N}x,\sqrt{N}y)\right)|\leq
Ce^{-Nc}.$ (8.77)

Employing (8.54) and (8.22) we see that same claim holds for
$D_{N}(x,y)-\mathcal{D}(\sqrt{N}x,\sqrt{N}y)$ and
$I_{N}(x,y)-\mathcal{I}(\sqrt{N}x,\sqrt{N}y)$, which suffices to demonstrate
the desired claim. ∎

Now that we have completed the even case of Lemma 8.3 we will now describe the
modifications that need to be made when $N$ is odd. As such, for the remainder
of this section we will assume that $N$ is odd. Although this case is not
treated in [29], it contains all of the ingredients to employ the formulas for
this case in [58, 28]. In particular, we see that the eigenvalue process for
$GEE(N,\tau)$ is of the form considered in (1.4) of [58] with

$w(z)=e^{-\frac{N}{2(1+\tau)}z^{2}}\sqrt{\mathrm{erfc}\left(|\mathrm{Im}(z)|\sqrt{\frac{2N}{1-\tau^{2}}}\right)}.$
(8.78)

In particular, we may employ the formulas they provide in their Section 7,
using the family of monic skew-orthogonal polynomials found in Theorem 1 of
[29]. The result of this computation will lead to roughly the same kernel as
in the even case, only with a few additional terms which we will show are
negligible with the same bounds as above.

On the other hand, due to a number of differences in notation between [29] and
[58], as well the difference in normalization between [29] and our own, we
will provide some expressions for how the functions defined in [29] equate to
the functions defined in [58].

In addition, to avoid confusion over the change in normalization between these
works and our own, during this translation, we will first compute the
corresponding kernel, $\bar{S}_{N}(x,y)$, defined with respect to the
$N$-independent measure

$\bar{w}(z)=e^{-\frac{1}{2(1+\tau)}z^{2}}\sqrt{\mathrm{erfc}\left(|\mathrm{Im}(z)|\sqrt{\frac{2}{1-\tau^{2}}}\right)}.$
(8.79)

By Theorem 1 of [29] we may apply Section 7 of [58] with

$\tilde{q}_{2i}(x)=e^{-\frac{x^{2}}{2(1+\tau)}}C_{2i}(x),\;\;\;\tilde{q}_{2i+1}(x)=e^{-\frac{x^{2}}{2(1+\tau)}}(C_{2i+1}(x)-2iC_{2i-1}(x)).$
(8.80) $2\tilde{\epsilon}q_{i}(x)=\int\mathrm{Sign}(x-t)\tilde{q}_{i}(t)dt.$
(8.81)

Employing Theorem 1 again and the formula (see [40])

$\int e^{-x^{2}}H_{2i}(xy)dx=\sqrt{\pi}\frac{(2m)!}{m!}(y^{2}-1)^{m},$ (8.82)

we see that the constants in the expression for $\bar{S}_{N}$ in [58] are
given by

$r_{i}=-2\sqrt{2\pi}(1+\tau)(2i)!,\;\;s_{2i+1}=0,\;\;s_{2i}=\sqrt{\pi(1+\tau)}\frac{(2i)!}{2^{i}(i!)}.$
(8.83)

Note the difference in the sign of $r_{i}$ here and in [29] which occurs due
to the sign difference between the skew-symmetric inner products of [29] and
[58]. With these relations we have that

$\bar{S}_{N}(y,x)=I+II+III,$ (8.84)
$I=\frac{-1}{2\sqrt{2\pi}(1+\tau)}\sum_{j=0}^{(N-3)/2}\frac{2}{(2i)!}\left(\tilde{q}_{2j}(y)\tilde{\epsilon}q_{2j+1}(x)-\tilde{q}_{2j+1}(y)\tilde{\epsilon}q_{2j}(x)\right),$
(8.85)
$II=-\frac{((N-1)/2)!2^{(N-1)/2}}{2\sqrt{2\pi}(1+\tau)(N-1)!}\sum_{j=0}^{(N-3)/2}\frac{2}{j!2^{j}}\left(\tilde{q}_{2j+1}(y)\tilde{\epsilon}q_{N-1}(x)-\tilde{q}_{N-1}(y)\tilde{\epsilon}q_{2j+1}(x)\right),$
(8.86)
$III=\frac{\tilde{q}_{N-1}(y)}{\sqrt{(1+\tau)\pi}}\frac{2^{N-1}((N-1)/2)!}{(N-1)!}=e^{-\frac{y^{2}}{2(1+\tau)}}\frac{C_{N-1}(y)}{\sqrt{(1+\tau)\pi}}\frac{2^{N-1}((N-1)/2)!}{(N-1)!}.$
(8.87)

We note that $I$ coincides with (4.5) of [29] after making the replacement
$N\mapsto N-1$. In particular, by their (5.24) we have that

$I=\frac{e^{-\frac{x^{2}+y^{2}}{2(1+\tau)}}}{\sqrt{2\pi}}\sum_{k=0}^{N-3}\frac{1}{k!}C_{k}(x)C_{k}(y)+\frac{e^{-\frac{y^{2}}{2(1+\tau)}}}{2\sqrt{2\pi}(1+\tau)}\frac{C_{N-2}(y)\Phi_{N-3}(x)}{(N-3)!}.$
(8.88)

We further note that (4.40) and the expression directly proceeding (5.10) in
[29] show that

$\tilde{\epsilon}q_{2i+1}(x)=-(1+\tau)e^{-\frac{x^{2}}{2(1+\tau)}}C_{2i}(x),\;\;\;\tilde{\epsilon}q_{2i+2}(x)-(2i+1)\tilde{\epsilon}q_{2i}(x)=-(1+\tau)e^{-\frac{x^{2}}{2(1+\tau)}}C_{2i+1}(x).$
(8.89)

Repeatedly applying these relations we see that

$II=\frac{-1}{2\sqrt{2\pi}(1+\tau)(N-2)!}e^{-\frac{y^{2}}{2(1+\tau)}}C_{N-2}(y)\Phi_{N-1}(x)+$
(8.90)
$\frac{((N-1)/2)!2^{(N-1)/2}}{2\sqrt{2\pi}(N-1)!}\sum_{j=0}^{(N-3)/2}\frac{e^{-\frac{x^{2}+y^{2}}{2(1+\tau)}}}{j!2^{j}}C_{N-1}(y)C_{2j}(x).$
(8.91)

Together these give a workable expression for $\bar{S}_{N}(y,x)$. They define
as well

$\bar{K}_{N}(y,x)=\begin{bmatrix}D\bar{S}_{N}(y,x)&\bar{S}_{N}(y,x)\\\
-\bar{S}_{N}(x,y)&\bar{S}_{N}I(y,x)+\frac{1}{2}\mathrm{Sign}(x-y)\\\
\end{bmatrix},$ (8.92)
$D\bar{S}_{N}(y,x)=\partial_{x}\bar{S}_{N}(y,x),\;\;\;\bar{S}_{N}I(y,x)=\frac{1}{2}\int\mathrm{Sign}(y-z)\bar{S}_{N}(z,x)dz=\int_{x}^{y}\bar{S}_{N}(z,x)dz.$
(8.93)

Note that there appears to be a typo in this expression in [58], as one sees
by comparison with their (5.3) which appears directly before it. We observe
that (8.92) differs in form from (8.53). This difference is not essential
though, and reflects the non-uniqueness of the kernel for a Pfaffian process.
In particular, we may instead take

$\bar{K}_{N}(y,x)=-\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\begin{bmatrix}D\bar{S}_{N}(y,x)&\bar{S}_{N}(y,x)\\\
-\bar{S}_{N}(y,x)&\bar{S}_{N}I(y,x)+\frac{1}{2}\mathrm{Sign}(x-y)\\\
\end{bmatrix}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}^{T}=$ (8.94)
$\begin{bmatrix}\bar{S}_{N}I(x,y)+\frac{1}{2}\mathrm{Sign}(y-x)&\bar{S}_{N}(x,y)\\\
-\bar{S}_{N}(y,x)&D\bar{S}_{N}(x,y)\\\ \end{bmatrix},$ (8.95)

as this operation does not affect the value of the Pfaffian. Thus finally,
restoring our normalization we see that when $N$ is odd, we may take
$K_{N}(x,y)$ defined by (8.53) and (8.54), as in the even case and
$S_{N}(x,y)$ defined by $\bar{S}_{N}(\sqrt{N}y,\sqrt{N}x)$

###### Proof of Lemma 8.3 with $N$ odd.

We note that the first term, $I$, which defines $\bar{S}_{N}(x,y)$ coincides
with the kernel with $N\mapsto N-1$. In particular, the proof of Lemma 8.3 in
the even case shows that for $x,y\in(-(1+\tau)\sqrt{N},(1+\tau)\sqrt{N})$
there is $c>0$ such that

$I=\bar{S}_{N}(x,y)+O(e^{-Nc}),$ (8.96)

and with the desired uniformity. In particular, we only need to show that the
terms $II$ and $III$ are negligible for
$x,y\in(-(1+\tau)\sqrt{N},(1+\tau)\sqrt{N})$. Repeatedly applying (8.69) and
(8.72) we see that there is $C,c>0$ such that

$|II|\leq
Ce^{-Nc}+Ce^{-Nc}N^{2}\sum_{j=0}^{(N-3)/2}\frac{\sqrt{(2j)!}}{j!2^{j}}\leq
C^{2}e^{-Nc/2},$ (8.97)

so that $II$ is negligible. A similar analysis of $III$ completes the proof. ∎

##  Appendix A Computation of Covariances and Conditional Laws

The primary purpose of this appendix will be to study the covariance structure
of the centered Gaussian vector

$(\zeta(\textbf{n}),\textbf{G}_{E}(\textbf{n}),\nabla_{E}\textbf{G}(\textbf{n}),\zeta(\textbf{n}(r)),\textbf{G}_{E}(\textbf{n}(r)),\nabla_{E}\textbf{G}(\textbf{n}(r))).$
(A.1)

The main result is given by Lemma A.1. The remainder of this section will then
consist of computing various conditional expectations and densities involving
(A.1), as well as establishing some basic results about them.

We define functions

$\Phi_{1}(x)=px^{p-1},\;\;\;\Phi_{2}(x)=\tau p(p-1)x^{p-2},$ (A.2)

so that

$\mathbb{E}[g_{i}(x)g_{j}(y)]=\delta_{ij}\Phi_{1}((x,y))+x_{j}y_{i}\Phi_{2}((x,y)).$
(A.3)

The proof we give for Lemma A.1 will actually hold for any centered Gaussian
random field with covariance given by (A.3), and with G and $\zeta$ defined
identically. In particular, it holds for the more general models considered in
[32, 38]. We believe in addition that this generality may assist in the
readability and verification of Lemma A.1.

We further define the function $\Phi_{3}(r)=\Phi_{1}(r)+r\Phi_{2}(r)$. With
$\delta$ denoting the standard Kronecker $\delta$, we will further denote
$\delta_{ijk}=\delta_{ij}\delta_{jk}$,
$\delta_{ijkl}=\delta_{ijk}\delta_{kl}$, and $\delta_{ij\neq
N-1}=\delta_{ij}(1-\delta_{i(N-1)})$. We will also use the short-hand
$x_{*}=\sqrt{1-x^{2}}$. Finally, we fix and omit the choice of $E$ given by
the following lemma from the notation.

######  Lemma A.1.

For any $r\in[-1,1]$, there exists a choice of orthonormal frame field
$E=(E_{i})_{i}$, such that for $1\leq i,j,\alpha,\beta\leq N-1$

$\mathbb{E}[G_{\alpha}(\textbf{n})G_{\beta}(\textbf{n}(r))]=\delta_{\alpha\beta\neq
N-1}\Phi_{1}(r)+\delta_{\alpha\beta(N-1)}(r\Phi_{1}(r)-r_{*}^{2}\Phi_{2}(r))$
(A.4)
$\mathbb{E}[G_{\alpha}(\textbf{n})\zeta(\textbf{n}(r))]=-\mathbb{E}[\zeta(\textbf{n})G_{\alpha}(\textbf{n}(r))]=\delta_{\alpha(N-1)}r_{*}\Phi_{3}(r)$
(A.5) $\mathbb{E}[\zeta(\textbf{n})\zeta(\textbf{n}(r))]=r\Phi_{3}(r)$ (A.6)
$\mathbb{E}[E_{i}G_{\alpha}(\textbf{n})G_{\beta}(\textbf{n}(r))]=-\mathbb{E}[G_{\beta}(\textbf{n})E_{i}G_{\alpha}(\textbf{n}(r))]=$
(A.7) $r_{*}[\delta_{\alpha\beta\neq
N-1}\delta_{i(N-1)}\Phi_{1}^{\prime}(r)+\delta_{\beta i\neq
N-1}\delta_{\alpha(N-1)}\Phi_{2}(r)+\delta_{\beta(N-1)}\delta_{\alpha
i}\Phi_{3}(r)$ (A.8) $\delta_{\alpha\beta
i(N-1)}(r\Phi_{3}^{\prime}(r)-\Phi_{2}^{\prime}(r))]$ (A.9)
$\mathbb{E}[E_{i}G_{\alpha}(\textbf{n})\zeta(\textbf{n}(r))]=\mathbb{E}[\zeta(\textbf{n})E_{i}G_{\alpha}(\textbf{n}(r))]=\delta_{\alpha
i(N-1)}r_{*}^{2}\Phi_{3}^{\prime}(r)-\delta_{\alpha i}r\Phi_{3}(r)$ (A.10)
$\mathbb{E}[E_{i}G_{\alpha}(\textbf{n})E_{j}G_{\beta}(\textbf{n}(r))]=r_{*}^{4}\delta_{ij\alpha\beta(N-1)}\Phi_{2}^{\prime\prime}(r)$
(A.11)
$-\Phi^{\prime}_{2}(r)r_{*}^{2}(\delta_{ij\alpha\beta(N-1)}5r+\delta_{i\alpha\neq(N-1)}\delta_{\beta
j(N-1)}r+\delta_{i\alpha(N-1)}\delta_{\beta j\neq(N-1)}r$ (A.12)
$+\delta_{\alpha j(N-1)}\delta_{i\beta\neq(N-1)}+\delta_{\alpha
j\neq(N-1)}\delta_{i\beta(N-1)}+\delta_{\alpha\beta(N-1)}\delta_{ij\neq(N-1)})$
(A.13)
$-r_{*}^{2}\Phi_{1}^{\prime\prime}(r)(\delta_{ij\alpha\beta(N-1)}r+\delta_{\alpha\beta\neq(N-1)}\delta_{ij(N-1)})$
(A.14) $-(\delta_{j\beta(N-1)}\delta_{\alpha i}+\delta_{j\beta}\delta_{\alpha
i(N-1)})r_{*}^{2}(\Phi_{2}(r)+\Phi_{1}^{\prime}(r))$ (A.15)
$+\Phi_{1}^{\prime}(r)(\delta_{ij\neq
N-1}+r\delta_{ij(N-1)})(\delta_{\alpha\beta\neq
N-1}+r\delta_{\alpha\beta(N-1)})$ (A.16) $+(\delta_{i\beta\neq
N-1}+r\delta_{i\beta(N-1)})(\delta_{\alpha j\neq N-1}+r\delta_{\alpha
j(N-1)})\Phi_{2}(r)+r\Phi_{3}(r)\delta_{i\alpha}\delta_{j\beta}.$ (A.17)

Before proceeding to the proof we observe that in the purely relaxational case
(i.e. $\Phi_{1}^{\prime}=\Phi_{2}$) we recover Lemma 54 of [13] with
$\Phi_{1}=\nu^{\prime}$ and $q_{1}=q_{2}=1$ (up to constant differences
resulting from their normalization $\nu(1)=1$), and in the case of $r=1$ we
recover the covariance structure described on pg. 13 of [38].

###### Proof.

We will take the orthonormal frame field constructed in the proof of Lemma 30
of [61], though we will need to recall its construction to establish the
precise properties of it that we will use.

We let $P_{N}:S^{N-1}\to\mathbb{R}^{N-1}$ denote projection given by omitting
the last coordinate. We let $R_{r,N}:S^{N-1}\to S^{N-1}$ denote the rotation
mapping

$R_{r,N}(x_{1},\dots x_{N})=(x_{1},\dots
x_{N-2},rx_{N-1}-r_{*}x_{N},r_{*}x_{N-1}+rx_{N}).$ (A.18)

We observe that $P_{N}(\textbf{n})=P_{N}\circ R_{r,N}(\textbf{n}(r))=0$. Then
taking small enough neighborhoods of n and $\textbf{n}(r)$ in $S^{N-1}$ that
the restrictions of $P_{N}$ and $P_{N}\circ R_{r,N}$ are diffeomorphisms onto
their image, we define $\hat{g}_{i}^{1}=g_{i}\circ P_{N}^{-1}$ and
$\hat{g}_{i}^{r}=g_{i}\circ(P_{N}\circ R_{r,N})^{-1}$, and similarly for
$\hat{\zeta}^{1}$ and $\hat{\zeta}^{r}.$

We now claim that there is an orthonormal frame field $E=(E_{i})_{i=1}^{N-1}$
such that for any $1\leq i,j\leq N-1$,

$(E_{i}\zeta(\textbf{n}),E_{i}G_{j}(\textbf{n}))=(\partial_{i}\hat{\zeta}^{1}(0),\partial_{i}\hat{g}_{j}^{1}(0)-\delta_{ij}\hat{g}^{1}_{N}(0))$
(A.19)

and such that for $1\leq j<N-1$ and $1\leq i\leq N-1$,

$(E_{i}\zeta(\textbf{n}(r)),E_{i}G_{j}(\textbf{n}(r)))=(\partial_{i}\hat{\zeta}^{r}(0),(\partial_{i}\hat{g}_{j}^{r})(0)-\delta_{ij}(r_{*}\hat{g}_{N-1}^{r}(0)+r\hat{g}_{N}^{r}(0))),$
(A.20)
$E_{i}G_{N-1}(\textbf{n}(r))=r\partial_{i}\hat{g}_{N-1}^{r}(0)-r_{*}\partial_{i}\hat{g}_{N}^{r}(0)-\delta_{i(N-1)}(r_{*}\hat{g}_{N-1}^{r}(0)+r\hat{g}_{N}^{r}(0)),$
(A.21)

where in all the expressions above, $\partial_{i}$ denotes the $i$-th standard
partial derivative on $\mathbb{R}^{N-1}$.

As stated above, we may infact take the orthonormal frame field constructed in
the proof of Lemma 30 in [61], or more precisely, in footnote 5 on their pg.
36. For clarity, we now recall it’s construction. Let
$(\frac{\partial}{\partial x_{i}})_{i=1}^{N-1}$ denote the pullback of the
standard frame field $(\frac{d}{dx_{i}})_{i=1}^{N-1}$ on $\mathbb{R}^{N-1}$ by
$P_{N}\circ R_{r,N}$. If we now denote the standard frame field on
$\mathbb{R}^{N}$ as $(\frac{\partial^{euc}}{\partial^{euc}x_{i}})_{i=1}^{N}$,
we see that with respect to the inclusion $T_{\textbf{n}(r)}S^{N-1}\subseteq
T_{\textbf{n}(r)}\mathbb{R}^{N}$ induced by the standard embedding
$S^{N-1}\subseteq\mathbb{R}^{N}$, we have that for $1\leq i<N-1$,

$\frac{\partial}{\partial
x_{i}}(\textbf{n}(r))=\frac{\partial^{euc}}{\partial^{euc}x_{i}},\;\;\;\;\;\frac{\partial}{\partial
x_{N-1}}(\textbf{n}(r))=r\frac{\partial^{euc}}{\partial^{euc}x_{N-1}}-r_{*}\frac{\partial^{euc}}{\partial^{euc}x_{N}},$
(A.22)

In particular, we observe that $(\frac{\partial}{\partial
x_{i}}(\textbf{n}(r)))_{i=1}^{N-1}$ is an orthonormal basis of
$T_{\textbf{n}(r)}S^{N-1}$. Thus we may form an orthonormal frame field on a
neighborhood of $\textbf{n}(r)$ by taking its value at any point $x$ to be
given by the parellel transport of $(\partial_{i})_{i=1}^{N-1}$ along the
unique geodesic of $S^{N-1}$ connecting $\textbf{n}(r)$ to $x$. If $r=1$, we
extend this orthonormal frame field arbitrarily to the remainder of the sphere
to obtain $E$. Otherwise, perform the same construction with $r=1$ to obtain a
field around n, and potentially shrinking our neighborhood, we may extend
these to obtain our desired $E$. We must now verify the claims that $E$
satisfies (A.19-A.21). We observe that by our construction (A.19) is a special
case of (A.20-A.21), so we are reduced to studying our frame field around
$\textbf{n}(r)$.

For this, we note that by routine computation, and $x\in\mathbb{R}^{N-1}$
sufficiently small we have that

$\frac{\partial}{\partial x_{i}}((P_{N}\circ
R_{r,N})^{-1}(x))=\partial_{i}^{euc}-x_{i}(r_{*}\partial_{N-1}^{euc}+r\partial_{N}^{euc})+O(\|x\|^{2}),$
(A.23) $\frac{\partial}{\partial x_{N-1}}((P_{N}\circ
R_{-\theta})^{-1}(x))=r\partial_{N-1}^{euc}-r_{*}\partial_{N}^{euc}-x_{N-1}(r_{*}\partial_{N-1}^{euc}+r\partial_{N}^{euc})+O(\|x\|^{2}),$
(A.24)

where we have abbreviated
$\partial^{euc}_{i}=\frac{\partial^{euc}}{\partial^{euc}x_{i}}$. Now working
in the coordinate system provided by $P_{N}\circ R_{r,N}$, we may write
$E_{i}=\sum_{j=1}^{N-1}a_{ij}(x)\frac{\partial}{\partial x_{i}}$ for some
smooth functions $a_{ij}$ defined on this neighborhood in $S^{N-1}$. It is
shown in [61] that for any $1\leq i,j,k\leq N-1$, $\frac{\partial}{\partial
x_{k}}a_{ij}(\textbf{n}(r))=0$, by using that the relevant Christoffel symbols
vanish at $\textbf{n}(r)$, and we note that by construction
$a_{ij}(\textbf{n}(r))=\delta_{ij}$. In particular, $E_{i}$ coincides with
$\frac{\partial}{\partial x_{i}}$ to first order in $x$, so that (A.23) and
(A.24) hold with $\frac{\partial}{\partial x_{i}}$ replaced by $E_{i}$.
Additionally, we see that

$E_{i}(G(x),E_{j}(x))|_{x=\textbf{n}(r)}=\sum_{k=1}^{N-1}\frac{\partial}{\partial
x_{i}}(G(x),a_{jk}(\textbf{n}(r))\frac{\partial}{\partial
x_{k}})\bigg{|}_{x=\textbf{n}(r)}=(E_{i}G(x),E_{j}(x))|_{x=\textbf{n}(r)}.$
(A.25)

Combining these results we obtain (A.20) and (A.21).

Now we observe that denoting

$\rho_{N}(x,y)=\sum_{i=1}^{N-2}x_{i}y_{i}+rx_{N-1}y_{N-1}+r_{*}x_{N-1}\|y\|_{*}-r_{*}y_{N-1}\|x\|_{*}+r\|x\|_{*}\|y\|_{*},$
(A.26)

that we have

$\mathbb{E}[\hat{g}_{i}^{1}(x)\hat{g}_{j}^{r}(y)]=\delta_{ij}\Phi_{1}(\rho_{N}(x,y))+(P_{N}^{-1}(x))_{j}((P_{N}\circ
R_{r,N})^{-1}(y))_{i}\Phi_{2}(\rho_{N}(x,y)),$ (A.27)

for $x,y\in\mathbb{R}^{N-1}$ such that the functions appearing on the left-
hand side are well defined. We observe similarly that for $1\leq i\leq N$ and
$x,y\in\mathbb{R}^{N-1}$ around zero,

$P_{N}^{-1}(x)_{i}=x_{i}\delta_{i\neq N}+\delta_{iN}+O(\|x\|^{2}),$ (A.28)
$(P_{N}\circ
R_{r,N})^{-1}(y)_{i}=y_{i}(\delta_{i\neq(N-1)}+r\delta_{i(N-1)})+\delta_{i(N-1)}r_{*}+\delta_{iN}(r-r_{*}y_{N-1})+O(\|y\|^{2}),$
(A.29)
$\rho_{N}(x,y)=\sum_{i=1}^{N-2}x_{i}y_{i}+rx_{N-1}y_{N-1}+r_{*}x_{N-1}-r_{*}y_{N-1}+r+O(\|x\|^{2}+\|y\|^{2}).$
(A.30)

We now simply recall eqn. 5.5.4 of [2], which states that for an arbitrary
centered Gaussian field, $h$, defined on an open subset of $\mathbb{R}^{m}$,
and with smooth covariance function $C$, we have that

$\mathbb{E}[\frac{d^{k}}{dx_{i_{1}}\dots
dx_{i_{k}}}h(x)\frac{d^{l}}{dy_{j_{1}}\dots
dy_{j_{l}}}h(y)]=\frac{d^{k}}{dx_{i_{1}}\dots
dx_{i_{k}}}\frac{d^{l}}{dy_{j_{1}}\dots dy_{j_{l}}}C(x,y).$ (A.31)

Using this formula, (A.19-A.21), and the above expressions for $x,y\sim 0$,
the lemma now follows from a long but routine computation. ∎

From this result, we observe the following corollary, which allows us to
partition the components of (A.1) into independent blocks.

######  Corollary A.2.

Let us take $r\in(-1,1)$, and the choice of orthonormal frame field
$(E_{i})_{i=1}^{N-1}$ given in Lemma A.1. Then all of the following random
variables are independent

  1. 1.

$(E_{i}G_{j}(\textbf{n}),E_{j}G_{i}(\textbf{n}),E_{i}G_{j}(\textbf{n}(r)),E_{j}G_{i}(\textbf{n}(r))),$
for $1\leq i<j<N-1$,

  2. 2.

$(E_{i}G_{i}(\textbf{n})+\zeta(\textbf{n}),E_{i}G_{i}(\textbf{n}(r))+\zeta(\textbf{n}(r))),$
for $1\leq i<N-1$,

  3. 3.

$(G_{i}(\textbf{n}),E_{N-1}G_{i}(\textbf{n}),E_{i}G_{N-1}(\textbf{n}),G_{i}(\textbf{n}(r)),E_{N-1}G_{i}(\textbf{n}(r)),E_{i}G_{N-1}(\textbf{n}(r)))$
for $1\leq i<N-1,$

  4. 4.

$(\zeta(\textbf{n}),\zeta(\textbf{n}(r)),G_{N-1}(\textbf{n}),G_{N-1}(\textbf{n}(r)),E_{N-1}G_{N-1}(\textbf{n}),E_{N-1}G_{N-1}(\textbf{n}(r))).$

We now go back to our original setting of (1.2). That is, we take

$\Phi_{1}(x)=px^{p-1},\;\;\Phi_{2}(x)=\tau
p(p-1)x^{p-2},\;\;\Phi_{3}(x)=p\alpha x^{p-1}.$ (A.32)

We now prepare to derive some statements involving the conditional densities.
To establish non-degeneracy in some cases, we will often use an observation
from (1.5), which shows that we may write

$\textbf{f}(x)=c_{s}\textbf{f}^{s}(x)+c_{a}\textbf{f}^{a}(x),$ (A.33)

where $c_{s},c_{a}$ are constants, and $\textbf{f}^{s}$ and $\textbf{f}^{a}$
are independent Gaussian vector-valued functions which satisfy (1.2) with
parameter $\tau=1$ and $\tau=-(p-1)^{-1}$ respectively. Abstractly, such a
decomposition is clear by noting that covariance function in (1.2) is linear
in $\tau$, but more explicitly we may write

$f^{s}_{i}(x)=\nabla^{euc}_{i}H(x),\;\;\;f^{a}_{i}(x)=\frac{1}{N}\sum_{k=1}^{N}x_{k}A_{ik}(x).$
(A.34)

We collect the facts we need about this decomposition in the following lemma,
which also verifies the equivalence of (1.2) and (1.5), as claimed in the
introduction.

######  Lemma A.3.

For $1\leq i,j\leq N$ and $x,y\in S_{N}$

$\mathbb{E}[f^{a}_{i}(x)f^{a}_{j}(y)]=\delta_{ij}(x,y)_{N}^{p-1}-\left(\frac{x_{j}y_{i}}{N}\right)(x,y)_{N}^{p-2},$
(A.35)
$\mathbb{E}[f^{s}_{i}(x)f^{s}_{j}(y)]=p\delta_{ij}(x,y)_{N}^{p-1}+p(p-1)\left(\frac{x_{j}y_{i}}{N}\right)(x,y)_{N}^{p-2}.$
(A.36)

In particular, the random vector-valued function defined by (1.5) satisfies
(1.2). Moreover, we have that $\lambda(x)=\frac{p}{N}H(x)$, so in particular
$\textbf{f}^{a}$ is independent of $\lambda$.

###### Proof.

We recall that for a spherical $\ell$-spin model, $\tilde{H}$, we have that
for $x,y\in S_{N}$

$\mathbb{E}[\tilde{H}(x)\tilde{H}(y)]=N(x,y)_{N}^{\ell}.$ (A.37)

In particular, by (A.31) above, we that

$\mathbb{E}[\nabla_{i}^{euc}H(x)\nabla_{j}^{euc}H(y)]=\frac{\partial^{2}}{\partial
x_{i}\partial y_{j}}N(x,y)_{N}^{p}$ (A.38)

which immediately gives (A.35). Similarly, we see that may compute that

$\mathbb{E}\left[\left(\frac{1}{N}\sum_{k=1}^{N}x_{k}A_{ik}(x)\right)\left(\frac{1}{N}\sum_{l=1}^{N}y_{l}A_{jl}(y)\right)\right]=\frac{1}{N}\sum_{k,l=1}^{N}x_{k}y_{l}(\delta_{ij}\delta_{kl}-\delta_{il}\delta_{jk})(x,y)_{N}^{p-2},$
(A.39)

which immediately gives (A.36). For the final claim we compute

$(x,\textbf{f}^{a}(x))_{N}=\frac{1}{N^{2}}\sum_{i,j=1}^{N}x_{i}x_{j}A_{ij}(x)=0,$
(A.40)
$(x,\textbf{f}^{s}(x))_{N}=\frac{1}{N}\sum_{i=1}^{N}x_{i}\nabla_{i}^{euc}H(x)=pH(x),$
(A.41)

where in the first line we have used that $A_{ij}$ and in the second we have
used that $H$ is a homogeneous polynomial of degree-$p$. ∎

We now proceed to the proof of Lemma 3.2. We will repeatedly use the formulas
for the conditional law of Gaussian distributions (see for example, (1.2.7)
and (1.2.8) of [2]), which reduces us to routine calculation using the
covariance computations in Lemma A.1.

###### Proof of Lemma 3.2.

We first observe that for $1\leq i<N-1$:

$\mathbb{E}[(E_{N-1}G_{i}(\textbf{n}(r)),E_{i}G_{N-1}(\textbf{n}(r)),G_{i}(\textbf{n}(r)))(E_{N-1}G_{i}(\textbf{n}),E_{i}G_{N-1}(\textbf{n}),G_{i}(\textbf{n}))^{t}]=$
(A.42) $p(p-1)\begin{bmatrix}r^{p-3}(r^{2}-(p-1)r_{*}^{2})&\tau
r^{p-3}(r^{2}-(p-1)r_{*}^{2})&-r^{p-2}r_{*}\\\ \tau
r^{p-3}(r^{2}-(p-1)r_{*}^{2})&r^{p-3}(r^{2}-\tau(p-1)r_{*}^{2})&-\tau
r^{p-2}r_{*}\\\ r^{p-2}r_{*}&\tau
r^{p-2}r_{*}&(p-1)^{-1}r^{p-1}\end{bmatrix},$ (A.43)
$\mathbb{E}[(G_{i}(\textbf{n}),G_{i}(\textbf{n}(r)))(G_{i}(\textbf{n}),G_{i}(\textbf{n}(r)))^{t}]=p\begin{bmatrix}1&r^{p-1}\\\
r^{p-1}&1\end{bmatrix}.$ (A.44)

In addition, we record the covariance matrix of the remaining special entries:

$\mathbb{E}[(\zeta(\textbf{n}(r)),G_{N-1}(\textbf{n}(r)),E_{N-1}G_{N-1}(\textbf{n}(r)))(\zeta(\textbf{n}),G_{N-1}(\textbf{n}),E_{N-1}G_{N-1}(\textbf{n}))^{t}]=$
(A.45)
$\begin{bmatrix}pr^{p}\alpha&pr_{*}r^{p-1}\alpha&-pr^{p-2}(1-pr_{*}^{2})\alpha\\\
-pr_{*}r^{p-1}\alpha&pr^{p-2}(r^{2}-r_{*}^{2}(p-1)\tau)&a_{1}(r)\\\
-pr^{p-2}(1-pr_{*}^{2})\alpha&-a_{1}(r)&a_{2}(r)\end{bmatrix}$ (A.46)

where here

$a_{1}(r)=r_{*}(r\Phi_{3}^{\prime}(r)+\Phi_{3}(r)-\Phi_{2}^{\prime}(r))=pr_{*}r^{p-3}(p\alpha
r^{2}-(p-1)(p-2)\tau)$ (A.47)
$a_{2}(r):=r_{*}^{2}[r_{*}^{2}\Phi_{2}^{\prime\prime}(r)-5r\Phi_{2}^{\prime}(r)-r\Phi_{1}^{\prime\prime}(r)-2(\Phi_{2}(r)+\Phi_{1}^{\prime}(r))]+(\Phi^{\prime}_{1}(r)+\Phi_{2}(r))r^{2}+r\Phi_{3}(r)=$
(A.48)
$r^{p-4}r_{*}^{4}p(p-1)(p-2)(p-3)\tau-r^{p-2}r_{*}^{2}p(p-1)(p+5p\tau-8\tau)+r^{p}p(p+2(p-1)\tau)$
(A.49)

From Corollary A.2 we have that for $1\leq i\leq N-1$, the random vectors
$(G_{i}(\textbf{n}),G_{i}(\textbf{n}(r)))$ are pairwise independent centered
Gaussian vectors, with covariance matrix given by the right-hand side of
(A.44) for $1\leq i<N-1$, and for $i=N-1$ (employing (A.46)) by

$\begin{bmatrix}p&pr^{p}-\tau p(p-1)r^{p-2}(1-r^{2})\\\ pr^{p}-\tau
p(p-1)r^{p-2}(1-r^{2})&p\end{bmatrix}.$ (A.50)

Thus we see that to verify the claims of Lemma 3.2 involving
$(G_{E}(\textbf{n}),G_{E}(\textbf{n}(r)))$ it suffices to show that the
matrices of (A.44) and (A.50) are invertible when $r\in(-1,1)$. The first is
obvious, so we focus on the second. We see that it suffices to verify that
with

$\hat{g}_{\tau}(r):=r^{p}-\tau(p-1)r^{p-2}(1-r^{2})$ (A.51)

we have that $|\hat{g}_{\tau}(r)|<1$ for $|r|<1$. As $\hat{g}_{\tau}(r)$ is
linear in $\tau$, it suffices to show that $|\hat{g}_{-(p-1)^{-1}}(r)|<1$ and
$|\hat{g}_{1}(r)|\leq 1$. Explicitly we have that
$\hat{g}_{-(p-1)^{-1}}(r)=r^{p-2}$, and that
$\hat{g}_{1}(r)=pr^{p}-(p-1)r^{p-2}$. The inequality for the first case is
obvious and for the second we note that for $r\geq 0$,

$r^{p}\leq pr^{p}-(p-1)r^{p}\hat{g}_{1}(r)\leq pr^{p-2}-(p-1)r^{p-2}=r^{p-2},$
(A.52)

and further noting that $\hat{g}_{1}(-r)=(-1)^{p}\hat{g}_{1}(r)$ established
the case. In particular, this confirms that the matrix (A.50) is invertible
for all $r\in(-1,1)$, and completes the first claims of Lemma 3.2.

To demonstrate the remaining claims in Lemma 3.2, we need to investigate the
conditional distribution of $(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))$. It is
clear that this is a centered Gaussian vector, so we only need to verify that
it is non-degenerate and compute its covariance.

To begin we will verify non-degeneracy. To do so let us recall the
decomposition $\textbf{g}=c_{s}\textbf{g}^{s}+c_{a}\textbf{g}^{a}$, from Lemma
A.3. The field $\textbf{g}^{a}$ is independent of $\zeta$, and so as
$c_{a}\neq 0$. We have just shown that
$(\textbf{G}^{a}(\textbf{n}),\textbf{G}^{a}(\textbf{n}(r)))$ is non-
degenerate, and independent of $\zeta$, we see that the conditional law of
$(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))$ is degenerate if and only if the
unconditional law of $(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))$ is non-
degenerate, which follows easily from (A.46).

We now set the notation

$\mathbb{E}_{\nabla,r}[-]=\mathbb{E}_{\nabla,r}[-|\nabla\textbf{G}(\textbf{n})=\nabla\textbf{G}(\textbf{n}(r))=0],$
(A.53)

which is valid for $r\in(-1,1)$ by the non-degeneracy established above. With
this notation we define

$\Sigma_{U}(r):=\mathbb{E}_{\nabla,r}[(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))^{t}].$
(A.54)

The remainder of the proof will be spent computing an expression for
$\Sigma_{U}(r)$, or more precisely $\Sigma_{U}(r)^{-1}$, which exists by the
non-degeneracy shown above.

To do this we denote the covariance matrix of the random vector

$(\frac{1}{\sqrt{p\alpha}}\zeta(\textbf{n}),\frac{1}{\sqrt{p}}G_{N-1}(\textbf{n}),\frac{1}{\sqrt{p\alpha}}\zeta(\textbf{n}(r)),\frac{1}{\sqrt{p}}G_{N-1}(\textbf{n}(r))),$
(A.55)

as $\bar{\Sigma}_{U}(r)$. This choice of normalization is choosen so that all
entries have unit variance. We note that $p\alpha\Sigma_{U}(r)^{-1}$ coincides
with the $2$-by-$2$ submatrix composed of the first and third rows and columns
of $\bar{\Sigma}_{U}(r)^{-1}$, so we will begin by computing this inverse.

If we denote

$c_{1}(r)=r^{p},\;\;c_{2}(r)=r_{*}r^{p-1}\sqrt{\alpha},\;\;c_{3}(r)=r^{p-2}(r^{2}-(p-1)\tau(1-r^{2})),$
(A.56)

then by (A.46) we have that

$C(r):=\begin{bmatrix}c_{1}(r)&c_{2}(r)\\\
-c_{2}(r)&c_{3}(r)\end{bmatrix},\;\;\bar{\Sigma}_{U}(r)=\begin{bmatrix}I&C(r)\\\
C(r)^{t}&I\\\ \end{bmatrix}.$ (A.57)

We will compute the block of inverse of this matrix by employing the Schur
complement formula, though to do this we will perform some preliminary
computations. We first compute that

$I-C(r)^{t}C(r)=\begin{bmatrix}f_{1}(r)&f_{2}(r)\\\
f_{2}(r)&f_{3}(r)\end{bmatrix},$ (A.58)

where here

$f_{1}(r):=1-r^{2p}-\alpha
r^{2p-2}(1-r^{2}),\;\;f_{2}(r):=-(p-1)\tau\sqrt{\alpha}r^{2p-3}(1-r^{2})^{3/2},$
(A.59)
$f_{3}(r):=1-r^{2p-2}(1-r^{2})\alpha-r^{2p-2}(r^{2}-(1-r^{2})(p-1)\tau)^{2}.$
(A.60)

We may then introduce functions

$b(r)=f_{1}(r)f_{3}(r)-f_{2}(r)^{2},\;\;k_{1}(r)=f_{3}(r),\;\;k_{2}(r)=c_{2}(r)f_{2}(r)-c_{1}(r)f_{3}(r),$
(A.61)

so that by the Schur complement formula

$\Sigma_{U}(r)^{-1}=\frac{1}{p\alpha b(r)}\begin{bmatrix}k_{1}(r)&k_{2}(r)\\\
k_{2}(r)&k_{1}(r)\end{bmatrix}.$ (A.62)

Finally by direct computation we see that

$b(r)=(1-r^{2p-2})^{2}-(p-1)^{2}\tau^{2}r^{2(p-2)}(1-r^{2})^{2},$ (A.63)
$k_{1}(r)=b(r)+r^{2p-2}(1-r^{2p-2}+\tau(p-1)(1-r^{2})),$ (A.64)
$k_{2}(r)=-r^{p}(1-r^{2p-2})-\tau(p-1)r^{3p-4}(1-r^{2}).$ (A.65)

∎

We now define the remaining conditional density matrices we need. As we only
need to understand their behavior as $r\to 1$ and $r\to 0$, we will avoid
giving as explicitly as we gave our expressions for $\Sigma_{U}$, writing them
instead as a product of certain fixed matrices.

To begin, we denote $\textbf{n}^{1}(r):=\textbf{n}$ and
$\textbf{n}^{2}(r):=\textbf{n}(r)$. We define for $k,l=1,2$ and $1\leq i<N-1$

$\Sigma_{Z}^{k,l}(r):=\frac{1}{p(p-1)}\mathbb{E}_{\nabla,r}[(E_{i}G_{N-1}(\textbf{n}^{k}(r)),E_{N-1}G_{i}(\textbf{n}^{k}(r)))(E_{i}G_{N-1}(\textbf{n}^{l}(r)),E_{N-1}G_{i}(\textbf{n}^{l}(r)))^{t}].$
(A.66)

We note that this quantity is independent of $i$, and that the conditional
covariance matrix of the vector

$\frac{1}{\sqrt{p(p-1)}}(E_{i}G_{N-1}(\textbf{n}),E_{N-1}G_{i}(\textbf{n}),E_{i}G_{N-1}(\textbf{n}(r)),E_{N-1}G_{i}(\textbf{n}(r)))$
(A.67)

on the event $(\textbf{G}(\textbf{n})=\textbf{G}(\textbf{n}(r))=0)$ is given
by the matrix

$\Sigma_{Z}(r):=\begin{bmatrix}\Sigma_{Z}^{1,1}(r)&\Sigma_{Z}^{1,2}(r)\\\
\Sigma_{Z}^{2,1}(r)&\Sigma_{Z}^{2,2}(r)\end{bmatrix}.$ (A.68)

We note that we also have that $\Sigma_{Z}^{1,1}(r)=\Sigma_{Z}^{2,2}(r)$ and
$\Sigma_{Z}^{1,2}(r)^{t}=\Sigma_{Z}^{2,1}(r)$. We note by Corollary A.2 this
is also the covariance matrix after conditioning on (A.80), as these random
variables are independent of $(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))$.

To give a more explicit representation, let us denote the unconditioned
covariance matrix of

$\frac{1}{\sqrt{p(p-1)}}(E_{i}G_{N-1}(\textbf{n}),E_{N-1}G_{i}(\textbf{n}),E_{i}G_{N-1}(\textbf{n}(r)),E_{N-1}G_{i}(\textbf{n}(r)))$
(A.69)

as $\Sigma_{A}(r)$ and further define

$\Sigma_{B}(r):=\frac{1}{p(p-1)}\mathbb{E}[(E_{i}G_{N-1}(\textbf{n}),E_{N-1}G_{i}(\textbf{n}),E_{i}G_{N-1}(\textbf{n}(r)),E_{N-1}G_{i}(\textbf{n}(r)))$
(A.70) $(G_{i}(\textbf{n}),G_{i}(\textbf{n}(r)))^{t}],$ (A.71)

then by Corollary A.2, (A.44), and the Schur complement formula we have that

$\Sigma_{Z}(r)=\Sigma_{A}(r)-\Sigma_{B}(r)\frac{1}{p(1-r^{2(p-1)})}\begin{bmatrix}1&-r^{p-1}\\\
-r^{p-1}&1\end{bmatrix}\Sigma_{B}(r)^{t}.$ (A.72)

Similarly if we denote the (unconditional) covariance matrix of

$(\zeta(\textbf{n}),\zeta(\textbf{n}(r)),G_{N-1}(\textbf{n}),G_{N-1}(\textbf{n}(r)))$
(A.73)

as $\Sigma_{C}(r)$ and

$\Sigma_{D}(r):=\frac{1}{p(p-1)}\mathbb{E}[(E_{N-1}G_{N-1}(\textbf{n})-\zeta(\textbf{n}),E_{N-1}G_{N-1}(\textbf{n}(r))-\zeta(\textbf{n}(r)))\times$
(A.74)
$(\zeta(\textbf{n}),\zeta(\textbf{n}(r)),G_{N-1}(\textbf{n}),G_{N-1}(\textbf{n}(r)))^{t}],$
(A.75)

we may express the covariance matrix of

$(E_{N-1}G_{N-1}(\textbf{n})-\zeta(\textbf{n}),E_{N-1}G_{N-1}(\textbf{n}(r)))-\zeta(\textbf{n}(r))$
(A.76)

conditioned on the event

$(\textbf{G}(\textbf{n})=\textbf{G}(\textbf{n}(r))=0,\zeta(\textbf{n})=u_{1},\zeta(\textbf{n}(r))=u_{2})$
(A.77)

as

$\Sigma_{S}(r)=\frac{1}{p(p-1)}\begin{bmatrix}p(p-1)(1+\tau)&a_{2}(r)\\\
a_{2}(r)&p(p-1)(1+\tau)\end{bmatrix}-\Sigma_{D}(r)\Sigma_{C}(r)^{-1}\Sigma_{D}(r)^{t},$
(A.78)

and the mean as

$m^{k}_{\ell}(r)=\Sigma_{D}(r)\Sigma_{C}(r)^{-1}e_{\ell}.$ (A.79)

We are now ready to give the proofs of Lemmas 3.3 and 3.4.

###### Proof of Lemma 3.3.

All of the independence statements follow from Corollary A.2. The law of the
elements of $(W^{k}_{N}(r),V^{k}_{N}(r))$ follows essentially by definition,
and as the distribution of $(G^{k}_{N}(r))_{k=1,2}$ is unaffected by the
conditioning, its law follows easily from Lemma A.1. ∎

###### Proof of Lemma 3.4.

With the results of Lemma 3.3, we need only show that for that for $k=1,2$ and
$r\in(-1,1)$, we have $[\Sigma_{S}(r)]_{kk}>0$. This is the conditional
variance of the matrix element $E_{N-1}G_{N-1}(\textbf{n})$ for $k=1$ and of
$E_{N-1}G_{N-1}(\textbf{n}(r))$ for $k=2$. We note that as these are the same,
it suffices to show the case of $k=1$. That is, we must show that
$E_{N-1}G_{N-1}(\textbf{n})$ is non-degenerate after conditioning on the event

$(\textbf{G}(\textbf{n})=\textbf{G}(\textbf{n}(r))=0,\zeta(\textbf{n})=u_{1},\zeta(\textbf{n}(r))=u_{2}).$
(A.80)

For this we will again employ Lemma A.3. In particular, as in the proof of
Lemma 3.2, we write $\textbf{g}=c_{s}\textbf{g}^{s}+c_{a}\textbf{g}^{a}$. As
$\textbf{g}^{a}$ is independent of $\zeta$, and $c_{s}\neq 0$, we see that it
is sufficient to show that $E_{N-1}G_{N-1}^{a}(\textbf{n})$ is non-degenerate
after only conditioning on the event
$(\textbf{G}^{a}(\textbf{n})=\textbf{G}^{a}(\textbf{n}(r))=0)$. The element
$E_{N-1}G_{N-1}^{a}(\textbf{n})$ is independent of all entries of
$(\textbf{G}^{a}(\textbf{n}),\textbf{G}^{a}(\textbf{n}(r)))$ except
$G_{N-1}^{a}(\textbf{n}(r))$. The covariance matrix of
$(G_{N-1}^{a}(\textbf{n}(r)),E_{N-1}G_{N-1}^{a}(\textbf{n}))$ is given by

$\begin{bmatrix}p&&p(p-2)r_{*}r^{p-3}\\\
p(p-2)r_{*}r^{p-3}&&p(p-2)\end{bmatrix},$ (A.81)

so it suffices to show that for $r\in(-1,1)$ that the determinant of this
matrix is positive, or equivalently that

$(p-2)(1-r^{2})r^{2(p-3)}<1.$ (A.82)

Employing (6.10) in the case of $2p-4$ we see that

$(p-2)(1-r^{2})r^{2(p-3)}\leq\left(\frac{p-2}{2p-5}\right)(1-r^{4p-10})\leq
1-r^{4p-10}<1$ (A.83)

where in the last two inequalities we have used that $p\geq 3$. ∎

Next we will give the proof of Lemma 5.3.

###### Proof of Lemma 5.3.

By employing (A.43) we see that $\Sigma_{B}(r)=O(r^{p-2})$ entrywise, so that

$\Sigma_{Z}(r)=\Sigma_{A}(r)+O(r^{2(p-2)}).$ (A.84)

Employing (A.44) we see that $\Sigma_{A}(r)=I+O(r^{p-3})$, so that in total we
have that $\Sigma_{Z}(r)=I+O(r^{p-3})$. In view of the expressions for
$Z_{N-1,i}^{k}(r)$ and $Z_{i,N-1}^{k}(r)$, this establishes the first two
claims in (5.74).

Employing (A.46) we see that
$\Sigma_{C}(r)=\mathrm{diag}(p\alpha,p\alpha,p,p)+O(r^{p-2})$, where
$\mathrm{diag}(x_{1},\dots,x_{n})$ denotes the diagonal matrix with entries
$x_{1},\dots x_{n}$. Noting that $a_{1}(r)=O(r^{p-3})$, we see that
$\Sigma_{D}(r)=O(r^{p-3})$, so that
$\Sigma_{D}(r)^{t}\Sigma_{C}(r)^{-1}\Sigma_{D}(r)=O(r^{2(p-3)})$. Now finally
$a_{2}(r)=O(r^{p-4})$ we see that

$\Sigma_{S}(r)=(1+\tau)I+O(r^{p-4}),$ (A.85)

which shows the first claim of (5.75), with the second following similarly. ∎

We will now finally prove a vanishing result for these functions as $r\to\pm
1$. For this, let us denote for $1\leq i\leq N-2$

$\sigma_{1}(r)=\mathbb{E}_{\nabla,r}[E_{N-1}G_{i}(\textbf{n})^{2}]=p(p-1)[\Sigma_{Z}(r)]_{22},$
(A.86)
$\sigma_{2}(r)=\mathbb{E}_{\nabla,r}[E_{N-1}G_{N-1}(\textbf{n})^{2}]=p(p-1)[\Sigma_{S}(r)]_{11}.$
(A.87)

######  Lemma A.4.

There is $C>0$ such that for $i=1,2$ and $r\in(-1,1)$

$\sigma_{i}(r)\leq C(1-r^{2}).$ (A.88)

###### Proof.

By the non-degeneracy established above, we see that $\sigma_{i}(r)>0$ for
$r\in(-1,1)$. As moreover, each $\sigma_{i}(r)$ is a rational function in $r$,
we see that it suffices to show that for $i=1,2$

$\lim_{r\to\pm 1}\sigma_{i}(r)=0.$ (A.89)

Moreover, as both expressions are even in $r$ it suffices to only consider the
case $r\to 1$.

For this we note that employing (A.43) the covariance matrix of the vector

$(E_{N-1}G_{i}(\textbf{n}),G_{i}(\textbf{n}(r)),G_{i}(\textbf{n}))$ (A.90)

is given by

$\begin{bmatrix}p(p-1)&p(p-1)r^{p-2}(1-r^{2})^{1/2}&0\\\
p(p-1)r^{p-2}(1-r^{2})^{1/2}&p&pr^{p-1}\\\ 0&pr^{p-1}&p\end{bmatrix}.$ (A.91)

By this and Corollary A.2, we have that

$\sigma_{1}(r)=p(p-1)-p^{2}(p-1)^{2}r^{2(p-2)}(1-r^{2})\begin{bmatrix}p&pr^{p-1}\\\
pr^{p-1}&p\end{bmatrix}^{-1}_{22}=$ (A.92)
$p(p-1)\left[1-\frac{(p-1)r^{2(p-2)}(1-r^{2})}{(1-r^{2p-2})}\right].$ (A.93)

Direct computation with the above expressions yield that

$\lim_{r\to 1}\sigma_{1}(r)=p(p-1)\left[1-(p-1)\lim_{r\to
1}\left(\frac{1-r^{2}}{1-r^{2p-2}}\right)\right]=0.$ (A.94)

The computation of $\sigma_{2}(r)$ proceeds similarly. Using (A.46) we see
that

$\sigma_{1}(r)=p(p+2\tau(p-1))-\frac{pa_{1}(r)^{2}}{p^{2}-(pr^{p}-\tau
p(p-1)r^{p-2}(1-r^{2}))^{2}}.$ (A.95)

We further note the following limits

$\lim_{r\to 1}a_{1}(r)/\sqrt{1-r^{2}}=p(p+2\tau(p-1)),$ (A.96) $\lim_{r\to
1}\frac{1-r^{2}}{p^{2}-(pr^{p}-\tau
p(p-1)r^{p-2}(1-r^{2}))^{2}}=\frac{1}{p^{2}(p+2\tau(p-1))}.$ (A.97)

Employing these one may similarly obtain that $\lim_{r\to 1}\sigma_{2}(r)=0$.
∎

##  Appendix B The Kac-Rice formula

In this section we derive the exact form of the Kac-Rice formula used in Lemma
3.1 above. This is essentially a modification of Theorem 6.4 of [7] to a
general Riemannian manifold. The method to adapt such a result from the
Euclidean case to a general Riemannian manifold is also discussed in (1.3.2)
of Chapter 6 in [7], though not in a generality sufficient for our case, we we
include a statement and proof of our result for completeness.

In preparation, we denote for a smooth manifold $M$, a vector field $f$ on
$M$, a function $h:M\to\mathbb{R}^{k}$, and a choice of
$B\subseteq\mathbb{R}^{k}$, the count

$N_{f,h}(B)=\\#\\{x\in M:f(x)=0,h(x)\in B\\}.$ (B.1)

######  Lemma B.1.

Let $M$ be a Riemannian manifold of dimension $d$, $f$ a random vector field
on $M$ and $h:M\to\mathbb{R}^{k}$ a random function such that $(f,h):M\to
TM\times\mathbb{R}^{k}$ is a smooth Gaussian random field. We assume that for
each $x\in M$, the Gaussian vector $f(x)$ is non-degenerate. Then for any
choice of a (piecewise) smooth orthonormal frame field $E=(E_{i})_{i=1}^{d}$
on $M$, and open $B\subseteq\mathbb{R}^{k}$, we have that

$\mathbb{E}[N_{f,h}(B)]=\int_{M}\mathbb{E}[|\det(\nabla_{E}f(x))|I(h(x)\in
B)|f_{E}(x)=0]\varphi_{f_{E}(x)}(0)\omega_{M}(dx),$ (B.2)

where $\omega_{M}$ denotes the Riemannian volume form on $M$,
$\varphi_{f_{E}(x)}(0)$ denotes the density of the random vector
$f_{i,E}(x)=(f(x),E_{i}(x))_{M}\in\mathbb{R}^{d}$, where $(*,*)_{M}$ denotes
the Riemannian metric on $M$, and lastly
$[\nabla_{E}f(x)]_{ij}=E_{i}f_{j,E}(x)$.

###### Proof.

We will begin with the case where $M$ is identified with an open subset $U$ of
$\mathbb{R}^{d}$, with the standard Euclidean metric, and where $E$ is the
standard Euclidean frame field. We first verify the conditions of Theorem 6.2
of [7] with $Z=f_{E}$. The only one which we have not taken as an assumption
is that

$\mathbb{P}\bigg{(}\text{there is }x\in U\text{ with }f(x)=0\text{ and
}\det(\nabla_{E}f(x))=0\bigg{)}=0.$ (B.3)

On the other hand employing their Proposition 6.5 (and an exhaustion of $U$ by
compact subsets) we see that (B.3) follows from the non-degeneracy assumption
on $f(x)$ for all $x\in U$. Thus by Theorem 6.4 of [7] we see that for any
bounded continuous function $g:\mathbb{R}^{k}\to\mathbb{R}$, we have that

$\mathbb{E}[\sum_{x\in
M:\nabla_{E}f(x)=0}g(h(x))]=\int_{U}\mathbb{E}[|\det(\nabla_{E}f(x))|g(h(x))|f_{E}(x)=0]\varphi_{f_{E}(x)}(0)\prod_{i=1}^{d}dx_{i}.$
(B.4)

On the other hand, as $B$ is open, we may write the indicator function of $B$
as the pointwise limit of an monotone increasing sequence of positive
continuous functions (take for example
$\epsilon^{-1}d(x,U^{c})/(1+\epsilon^{-1}d(x,U^{c}))$ where here
$d(x,A)=\inf_{y\in A}\|x-y\|$). By applying the monotone convergence theorem
to (B.4) for these functions we obtain (B.2) in our current case.

To proceed, we now relax the assumption on the metric and frame field. We will
denote by $\nabla^{euc}$ the standard Euclidean frame field on
$\mathbb{R}^{d}$, and denote by $\textbf{g}^{euc}$ and $\nabla^{euc}f$ the
quantities with respect to this basis. We observe that for $1\leq i\leq d$ we
have that $E_{i}(x)=[g(x)^{-1/2}\nabla^{euc}]_{i}$ where
$[g(x)]_{ij}=(E_{i}(x),E_{j}(x))$ is the metric tensor. We also have that
$f_{E}(x)=g(x)^{-1/2}\textbf{g}^{euc}(x)$ and that

$\nabla_{E}f(x)=g(x)^{-1}\nabla^{euc}f(x)+g^{-1/2}(x)(\nabla^{euc}g^{-1/2}(x))\textbf{g}^{euc}(x).$
(B.5)

From these relations we derive that
$\varphi_{f_{E}(x)}(0)=\det(g(x))^{1/2}\varphi_{\textbf{g}^{euc}(x)}(0)$ and

$\mathbb{E}[|\det(\nabla_{E}f(x))|I(f(x)\in B)|f_{E}(x)=0]=$ (B.6)
$\det(g(x))^{-1}\mathbb{E}[|\det(\nabla^{euc}f(x))|I(f(x)\in
B)|\textbf{g}^{euc}(x)=0].$ (B.7)

As we have that $\omega_{U}(dx)=\det(g(x))^{1/2}\prod_{i=1}^{d}dx_{i}$, we see
thus that the integral on the right-hand side of (B.4) is independent of both
the choice of metric on $M$, and of the choice of orthonormal frame. Thus we
have established the case where $M$ may be embedded as an open subset of
Euclidean space.

Now finally for the case of a general manifold $M$, we take a countable cover
of $M$ by Euclidean neighborhoods $(U_{i})_{i=1}^{\infty}$ and define
$V_{m}=\bigcup_{i=1}^{m}U_{i}$. We observe that all non-empty finite
intersections of the $U_{i}$ may be realized as open subsets of
$\mathbb{R}^{N}$, so by the inclusion-exclusion principle, we see that (B.2)
holds when both sides are restricted to $V_{m}$. Again applying the monotone
convergence theorem completes the proof. ∎

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