#  YITP-23-131
Real eigenvector distributions of random tensors
with backgrounds and random deviations

Naoki<EMAIL_ADDRESS>
Yukawa Institute for Theoretical Physics, Kyoto University,
and
CGPQI, Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

As in random matrix theories, eigenvector/value distributions are interesting
quantities of random tensors in their applications. Recently, real
eigenvector/value distributions of Gaussian random tensors have been
explicitly computed by expressing them as partition functions of quantum field
theories with quartic interactions. This procedure to compute distributions in
random tensors is general, powerful and intuitive, because one can take
advantage of well-developed techniques and knowledge of quantum field
theories. In this paper we extend the procedure to the cases that random
tensors have mean backgrounds and eigenvector equations have random
deviations. In particular, we study in detail the case that the background is
a rank-one tensor, namely, the case of a spiked tensor. We discuss the
condition under which the background rank-one tensor has a visible peak in the
eigenvector distribution. We obtain a threshold value, which agrees with a
previous result in the literature.

##  1 Introduction

Eigenvalue distributions are important quantities in random matrix models. The
most well known is the Wigner semi-circle law of the eigenvalue distribution,
which models energy spectra of strongly interacting many-body systems [1].
Eigenvalue distributions are also used as an important technique in solving
matrix models [2, 3]. Topological changes of eigenvalue distributions provide
insights into the QCD dynamics [4, 5].

It would be natural to ask how such knowledge about random matrices can be
generalized to random tensors. Random tensor models [6, 7, 8, 9] were
originally introduced to extend random matrix models, which are successful as
two-dimensional quantum gravity, to higher dimensional quantum gravity.
Recently random tensor models also play interesting roles in various other
subjects (See for instance [10]). While physically interesting matrices like
hermitian can be one-to-one mapped to sets of eigenvalues by symmetry
transformations, this cannot be done in general for tensors. However, we
sometimes encounter what we may call tensor eigenvectors/values [11, 12, 13,
14] in studies. A well-known example is the distribution of the energy spectra
of the spherical $p$-spin model [15, 16] for spin glasses, which was
comprehensively analyzed in [17]. In fact this is the same problem as
obtaining the real eigenvalue222More precisely, they are Z-eigenvalues in the
terminology of [11, 14]. distribution of a real symmetric random tensor.
Tensor eigenvector/value problems also appear in other contexts, such as
AdS/CFT [18], classical gravitational systems [19], and applied mathematics
for technologies [14].

Considering its wide appearance, it is worth effort to systematically
understand properties of tensor eigenvectors/values. Our focus is on their
distributions for Gaussian random tensors. Some interesting results have
already been obtained in the literature. In [20, 21] the expectation values of
numbers of real eigenvalues of random tensors were computed. In [22] the
maximum eigenvalues of random tensors were estimated in the large-$N$
limit333Throughout this paper, $N$ denotes the range of indices of tensors,
namely, an index takes values, $1,2,\cdots,N$.. In [23], the Wigner semi-
circle law was extended to a form for random tensors. In [24, 25, 26] the
present author computed real eigenvalue distributions of random tensors by
quantum field theoretical methods.

In the last works above by the present author, the procedure is to first
rewrite the eigenvector problems to partition functions of quantum field
theories with quartic interactions, and then compute the partition functions.
There are some good merits in this procedure; it is general, powerful, and
intuitive. As far as tensors have Gaussian distributions, one can in principle
extend the procedure to obtain quantum field theories of quartic interactions
for a wide range of other tensor problems, such as complex eigenvalue/vector
distributions, tensor rank decompositions, etc. Then, once such quantum field
theories have been obtained, one can use various well-developed quantum field
theoretical techniques, such as Schwinger-Dyson equations as in [25], etc.
Moreover, it is generally more intuitive to compute partition functions than
to directly treat systems of eigenvector/value polynomial equations. For
instance, in the large-$N$ analysis of [25], there exists a phase transition
point between perturbative and non-perturbative regimes of the quantum field
theory, and this point corresponds to the edge of the eigenvalue distribution.

The purpose of the present paper is to apply this quantum field theoretical
procedure to a slightly different setup than the previous works [24, 25, 26].
We assume the random tensors have mean values, namely, backgrounds. This is a
useful setup in the research of data analysis, in which backgrounds are
signals and deviations around them are noises [27]. It is an important
question under what conditions signals can be recovered from data contaminated
by noises [28, 29]. We also introduce random deviations to eigenvector
equations444This particular case will also be analyzed in detail in [30]..
This simulates solving approximately eigenvector equations, for instance, by
the Monte Carlo method or simulated annealing. As we will see, also in this
generalized setup, the distributions can be rewritten as partition functions
of quantum field theories with quartic interactions, and the partition
functions can be computed explicitly, even exactly in some cases.

This paper is organized as follows. In Section 2, we introduce a real
eigenvector equation with a tensor mean background and deviations to the
equation, and obtain an integral expression of the eigenvector distribution.
In Section 3, we derive the quantum field theory expressing a “signed”
distribution of the eigenvectors. This distribution is not authentic but is
weighted with an extra sign associated to each eigenvector. This distribution
is easier to compute, because the quantum field theory contains only a pair of
fermions. In particular, when the background is taken to be a rank-one tensor
(a spiked tensor), we obtain an exact expression of the distribution in terms
of hypergeometric functions. In Section 4 we derive the quantum field theory
expression of the (authentic) distribution of the eigenvectors. In particular
we explicitly derive the distribution for the spiked tensor case by using an
approximation taking advantage of the quantum field theoretical expression. In
Section 5, we compare the expressions of the distributions obtained in the
previous sections with Monte Carlo simulations. We obtain very good agreement,
including for the case treated by the approximation. In Section 6, we consider
the large-$N$ limit, especially paying attention to whether the rank-one
tensor background has a visible peak in the distributions. We derive the
scaling and the range of parameters in which this happens. The threshold value
is shown to agree with that of [29]. The last section is devoted to summary
and future prospects.

##  2 Real tensor eigenvector equation with backgrounds and deviations

In this paper we restrict ourselves to order-three tensors555Namely, tensors
have three indices. for simplicity. We consider the following eigenvector
equation [11, 12, 13, 14] with a background tensor $Q$ and a deviation vector
$\eta$,

$\displaystyle(Q_{abc}+C_{abc})v_{b}v_{c}=v_{a}+\eta_{a}.$ (1)

Here the indices take $a,b,c=1,2,\ldots,N$, and repeated indices are assumed
be summed over unless otherwise stated throughout this paper. We assume that
$Q,C$ are real symmetric order-three tensors and $v,\eta$ are real vectors:

$\displaystyle\begin{split}&Q_{abc}=Q_{bac}=Q_{bca}\in\mathbb{R},\\\
&C_{abc}=C_{bac}=C_{bca}\in\mathbb{R},\\\ &v_{a},\
\eta_{a}\in\mathbb{R}.\end{split}$ (2)

While $Q$ is an externally given background tensor, $C_{abc}$ is a random
tensor with Gaussian distribution of a zero mean value. The vector $\eta$
describes a deviation of the eigenvector equation, and is a random real vector
with Gaussian distribution of a zero mean value. We will compute the
distributions of $v$, namely the distributions of the real “eigenvector”
solutions to (1). Note that, if we ignore the background $Q$ and the deviation
$\eta$, the setup goes back to the cases previously studied in [24, 25, 26].

For given $Q,C,\eta$, the distribution of $v$ is given by

$\displaystyle\begin{split}\rho(v,Q,C,\eta)&=\sum_{i=1}^{\\#{\rm
sol}(Q,C,\eta)}\prod_{a=1}^{N}\delta(v_{a}-v_{a}^{i})\\\ &=\left|\det
M(v,Q,C)\right|\prod_{a=1}^{N}\delta\left(v_{a}+\eta_{a}-(Q_{abc}+C_{abc})v_{b}v_{c}\right)\end{split}$
(3)

where $v^{i}\ (i=1,2,\ldots,\\#{\rm sol}(Q,C,\eta))$ are all the real
solutions to (1), and $|\det M(v,Q,C)|$ is the absolute value of the
determinant of the matrix,

$\displaystyle M(v,Q,C)_{ab}=\frac{\partial}{\partial
v_{a}}\left(v_{b}+\eta_{b}-(Q_{bcd}+C_{bcd})v_{c}v_{d}\right)=\delta_{ab}-2(Q_{abc}+C_{abc})v_{c},$
(4)

which is the Jacobian factor associated to the change of the variables of the
delta functions in (3).

When $C,\eta$ have Gaussian distributions with zero mean values, the
eigenvector distributions are computed by taking the average over $C,\eta$:

$\displaystyle\begin{split}\rho(v,Q,\beta)&=\left\langle\rho(v,Q,C,\eta)\right\rangle_{C,\eta}\\\
&=\frac{1}{AA^{\prime}}\int_{\mathbb{R}^{\\#C}}dC\int_{\mathbb{R}^{N}}d\eta\,e^{-\alpha
C^{2}-\frac{1}{4\beta}\eta^{2}}\left|\det
M(v,Q,C)\right|\prod_{a=1}^{N}\delta\left(v_{a}+\eta_{a}-(Q_{abc}+C_{abc})v_{b}v_{c}\right)\end{split}$
(5)

where $\alpha,\beta>0$, $\\#C$ is the number666More explicitly,
$\\#C=N(N+1)(N+2)/6$. of the independent components of $C$,
$C^{2}=C_{abc}C_{abc}$, $\eta^{2}=\eta_{a}\eta_{a}$,
$A=\int_{\mathbb{R}^{\\#C}}dC\,e^{-\alpha C^{2}}$, and
$A^{\prime}=\int_{\mathbb{R}^{N}}d\eta\,e^{-\frac{1}{4\beta}\eta^{2}}$. Here a
slightly complicated introduction of $\beta$ is for later convenience. By
using the well known formula,
$\frac{1}{2\pi}\int_{\mathbb{R}}d\lambda\,e^{i\lambda x}=\delta(x)$, the
integration of the delta functions over $\eta$ in (5) can be rewritten as

$\displaystyle\frac{1}{A^{\prime}}\int_{\mathbb{R}^{N}}d\eta\,e^{-\frac{1}{4\beta}\eta^{2}}\prod_{a=1}^{N}\delta\left(v_{a}+\eta_{a}-(Q_{abc}+C_{abc})v_{b}v_{c}\right)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}d\lambda\,e^{-\beta\lambda^{2}+i\lambda_{a}\left(v_{a}-(Q_{abc}+C_{abc})v_{b}v_{c}\right)}.$
(6)

Therefore, by putting this into (5), we obtain

$\displaystyle\rho(v,Q,\beta)=\frac{1}{(2\pi)^{N}A}\int_{\mathbb{R}^{\\#C}}dC\int_{\mathbb{R}^{N}}d\lambda\
\left|\det M(v,Q,C)\right|e^{-\alpha
C^{2}-\beta\lambda^{2}+i\lambda_{a}(v_{a}-(Q_{abc}+C_{abc})v_{b}v_{c})}.$ (7)

The part $\left|\det M(v,Q,C)\right|$ in (7) needs a special care, because
taking an absolute value is not an analytic function. In Section 3, we will
consider the case that we ignore taking the absolute value. This makes the
problem easier and treatable by introducing only a pair of fermions, but is
still non-trivial and interesting. In Section 4, we will fully treat (7) by
introducing both bosons and fermions.

##  3 Signed distributions

###  3.1 Quantum field theory expression

The quantity we will compute in this section is defined by ignoring taking the
absolute value in (7):

$\displaystyle\rho^{\rm
signed}(v,Q,\beta)=\frac{1}{(2\pi)^{N}A}\int_{\mathbb{R}^{\\#C}}dC\int_{\mathbb{R^{N}}}d\lambda\det\\!M(v,Q,C)\,e^{-\alpha
C^{2}-\beta\lambda^{2}+i\lambda_{a}(v_{a}-(Q_{abc}+C_{abc})v_{b}v_{c})}.$ (8)

Following backward the derivation in Section 2, the distribution corresponds
to a “signed” distribution,

$\displaystyle\begin{split}\rho^{\rm signed}(v,Q,C,\eta)&=\sum_{i=1}^{\\#{\rm
sol}(Q,C,\eta)}{\rm
sign}\left(\det\\!M(v^{i},Q,C)\right)\prod_{a=1}^{N}\delta(v_{a}-v_{a}^{i}),\end{split}$
(9)

which has an extra sign of $\det\\!M(v^{i},Q,C)$ dependent on each solution
$v^{i}$, compared with (3). Note that the quantity (8) is a generalization of
the signed distribution computed in [24] to the case with backgrounds and
deviations. Though the quantity has no clear connections to (7), it provides a
simpler playground, and we will obtain an exact final expression with the
confluent hypergeometric functions of the second kind (or hermite
polynomials).

The determinant factor in (8) can easily be rewritten in a quantum field
theoretical form by introducing a fermion pair, $\bar{\psi}_{a},\psi_{a}\
(a=1,2,\cdots,N)$; $\det M=\int d\bar{\psi}d\psi\,e^{\bar{\psi}M\psi}$ [31].
This technique to incorporate determinants in quantum field theories is common
in treating disordered systems in statistical physics777See for instance [32]
and references therein.. Then (8) can be rewritten as

$\displaystyle\rho^{\rm
signed}(v,Q,\beta)=\frac{1}{(2\pi)^{N}A}\int_{\mathbb{R}^{\\#C}}dC\int_{\mathbb{R}^{N}}d\lambda\int
d\bar{\psi}d\psi\,e^{S^{\rm signed}_{\rm bare}},$ (10)

where

$\displaystyle S^{\rm signed}_{\rm bare}=-\alpha
C^{2}-\beta\lambda^{2}+i\lambda_{a}(v_{a}-(Q_{abc}+C_{abc})v_{b}v_{c})+\bar{\psi}_{a}\left(\delta_{ab}-2(Q_{abc}+C_{abc})v_{c}\right)\psi_{b}.$
(11)

Since $C$ and $\lambda$ appear at most quadratically in (11), they can be
integrated out by Gaussian integrations. We will first integrate over $C$ and
then over $\lambda$. Though the integrations are straightforward, the actual
computation is a little cumbersome, because of the anti-commuting nature of
the fermions and the necessity of symmetrization for the indices of $C_{abc}$.
However, we can take a shortcut by taking some results from [24], where there
are no $Q$ or $\eta$. Now, new terms in $S^{\rm signed}_{\rm bare}$ compared
to [24] are those depending on $Q$ and $\beta$, and are explicitly given by

$\displaystyle S_{\rm new}^{\rm
signed}=-\beta\lambda^{2}-i\lambda_{a}Q_{abc}v_{b}v_{c}-2Q_{abc}\bar{\psi}_{a}\psi_{b}v_{c}.$
(12)

Since the new terms do not contain $C$, the integration over $C$ proceeds in
the same way as in [24]. This integration cancels the overall factor $A^{-1}$
in (10), and also generates various terms being added to the action.
Collecting the terms depending on $\lambda$ among the generated ones,
$i\lambda_{a}v_{a}$ in (11), and the terms depending on $\lambda$ in (12), we
obtain the $\lambda$-dependent part of the action as

$\displaystyle S_{\lambda}^{\rm
signed}=-\frac{v^{4}}{12\alpha}B_{ab}\lambda_{a}\lambda_{b}+i\lambda_{a}(v_{a}+D^{\rm
signed}_{a}-D^{Q}_{a}),$ (13)

where $D^{Q}_{a}=Q_{abc}v_{b}v_{c}$, and $D^{\rm signed}$ can be taken from
[24]888$v_{a}+D_{a}^{\rm signed}$ corresponds to $D_{a}$ of [24].,

$\displaystyle D^{\rm signed}_{a}$
$\displaystyle=\frac{1}{3\alpha}\left(\bar{\psi}_{a}\,\psi\cdot
v\,v^{2}+\bar{\psi}\cdot v\,\psi_{a}\,v^{2}+\bar{\psi}\cdot v\,\psi\cdot
v\,v_{a}\right).$ (14)

Here we very frequently use an abusive notation $v^{p}:=|v|^{p}$ for
simplicity throughout this paper, since whether $v$ means vector or scalar
quantities are always obvious from contexts. The matrix $B$ is given by

$\displaystyle
B=3\left(1+\frac{4\alpha\beta}{v^{4}}\right)I_{\parallel}+\left(1+\frac{12\alpha\beta}{v^{4}}\right)I_{\perp},$
(15)

where $I_{\parallel}$ and $I_{\perp}$ are the projection matrices to the
parallel and the transverse subspaces against $v$:
$I_{\parallel}{}_{ab}=v_{a}v_{b}/v^{2},\
I_{\perp}{}_{ab}=\delta_{ab}-v_{a}v_{b}/v^{2}$. Then the integration over
$\lambda$ with the action (13) generates an action,

$\displaystyle\begin{split}\delta S^{\rm
signed}_{\lambda}=&-\frac{N}{2}\log\frac{v^{4}}{12\pi\alpha}-\frac{1}{2}\log\det
B\\\ &-\frac{3\alpha}{v^{4}}\left((v_{a}+D^{\rm
signed}_{a})B^{-1}_{ab}(v_{b}+D_{b}^{\rm signed})-2(v_{a}+D_{a}^{\rm
signed})B_{ab}^{-1}D^{Q}_{b}+D^{Q}_{a}B^{-1}_{ab}D^{Q}_{b}\right),\end{split}$
(16)

where the inverse of $B$ is given by

$\displaystyle B^{-1}=\frac{b_{\parallel}}{3}I_{\parallel}+b_{\perp}I_{\perp}$
(17)

with

$\displaystyle\begin{split}&b_{\parallel}=\frac{v^{4}}{v^{4}+4\alpha\beta},\\\
&b_{\perp}=\frac{v^{4}}{v^{4}+12\alpha\beta}.\end{split}$ (18)

When we consider the case with $Q=\beta=0$, the distribution (10) should agree
with the previous result of [24]. Therefore it is enough for us to compute the
additional part which appears only when $Q\neq 0$ or $\beta\neq 0$. By
subtracting $\delta S^{\rm signed}_{\lambda}$ for $Q=\beta=0$ in (16) and
using (17), we obtain

$\displaystyle\begin{split}&\delta S^{\rm signed}_{\lambda}-\delta S^{\rm
signed}_{\lambda}(Q=\beta=0)=\frac{1}{2}\log b_{\parallel}+\frac{N-1}{2}\log
b_{\perp}-\frac{3\alpha}{v^{4}}\bigg{[}\frac{b_{\parallel}-1}{3}(v+D^{\rm
signed}_{\parallel})^{2}\\\ &+(b_{\perp}-1)D^{\rm signed}_{\perp}\cdot D^{\rm
signed}_{\perp}-\frac{2b_{\parallel}}{3}(v+D^{\rm
signed}_{\parallel})D^{Q}_{\parallel}-2b_{\perp}D^{\rm signed}_{\perp}\cdot
D^{Q}_{\perp}+\frac{b_{\parallel}}{3}D^{Q}_{\parallel}{}^{2}+b_{\perp}D^{Q}_{\perp}\cdot
D^{Q}_{\perp}\bigg{]},\end{split}$ (19)

where $D^{\rm signed}_{\parallel}=v\cdot D^{\rm signed}/|v|,\ D^{\rm
signed}_{\perp}=I_{\perp}D^{\rm signed},\ D^{Q}_{\parallel}=v\cdot D^{Q}/|v|,\
D^{Q}_{\perp}=I_{\perp}D^{Q}$.

The previous result in [24] is given by

$\displaystyle\rho^{\rm
signed}(v,Q=0,\beta=0)=3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}\int
d\bar{\psi}d\psi\,e^{S_{\bar{\psi}\psi}},$ (20)

where

$\displaystyle S_{\bar{\psi}\psi}=-\frac{\alpha}{v^{2}}-2N\log
v+{\bar{\psi}_{\perp}}\cdot\psi_{\perp}-{\bar{\psi}_{\parallel}}{\psi_{\parallel}}-\frac{v^{2}}{6\alpha}\left({\bar{\psi}_{\perp}}\cdot\psi_{\perp}\right)^{2}.$
(21)

Adding (19) and the last term in (12) to (21) and doing some straightforward
computations, we finally obtain

$\displaystyle\begin{split}\rho^{\rm
signed}(v,Q,\beta)=&3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}(v^{4}+4\alpha\beta)^{-\frac{1}{2}}(v^{4}+12\alpha\beta)^{-\frac{N-1}{2}}\exp\left[-\frac{\alpha
v^{2}}{v^{4}+4\alpha\beta}\right]\\\ &\cdot\exp\left[\frac{2\alpha
b_{\parallel}vD^{Q}_{\parallel}-\alpha
b_{\parallel}(D^{Q}_{\parallel})^{2}-3\alpha b_{\perp}D^{Q}_{\perp}\cdot
D^{Q}_{\perp}}{v^{4}}\right]\int d\bar{\psi}d\psi\,e^{S^{\rm
signed}},\end{split}$ (22)

where

$\displaystyle\begin{split}S^{\rm
signed}&=\left(-2b_{\parallel}+1+\frac{2b_{\parallel}D^{Q}_{\parallel}}{v}\right){\bar{\psi}_{\parallel}}{\psi_{\parallel}}+\frac{2b_{\perp}}{v}D^{Q}_{\perp}\cdot\left({\bar{\psi}_{\perp}}{\psi_{\parallel}}+{\bar{\psi}_{\parallel}}\psi_{\perp}\right)+{\bar{\psi}_{\perp}}\cdot\psi_{\perp}\\\
&\ \
-2Q_{abc}\bar{\psi}_{a}\psi_{b}v_{c}+\frac{2v^{2}(b_{\perp}-1)}{3\alpha}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}{\bar{\psi}_{\perp}}\cdot\psi_{\perp}-\frac{v^{2}}{6\alpha}\left({\bar{\psi}_{\perp}}\cdot\psi_{\perp}\right)^{2}\end{split}$
(23)

with ${\psi_{\parallel}}=v\cdot\psi/|v|,\ \psi_{\perp}=I_{\perp}\psi$, etc.
Some details of the derivation are explained in Appendix A.

###  3.2 Rank-one $Q$

To study the formula (22) with (23) more explicitly, let us consider the case
that $Q$ is a rank-one tensor,

$\displaystyle Q_{abc}=q\,n_{a}n_{b}n_{c},$ (24)

where $q$ is real and $n$ is a normalized real vector ($|n|=1$). This is a
setup called a spiked tensor [27].

In the general situation, the vector $n$ is a linear combination of $v$ and
another vector $n_{1}$, which is a normalized vector transverse to $v$. Then
the transverse subspace to $v$ can further be divided into the subspace
parallel to $n_{1}$ and the $N-2$-dimensional subspace transverse to both $v$
and $n_{1}$. We denote the projector to the latter by $I_{\perp_{2}}$. Then
the transverse fermions, ${\bar{\psi}_{\perp}},\psi_{\perp}$, can further be
decomposed into ${\bar{\psi}_{\perp_{1}}}=n_{1}\cdot\bar{\psi}$ and
${\bar{\psi}_{\perp_{2}}}=I_{\perp_{2}}\bar{\psi}$ and similarly for
$\psi_{\perp}$. Note that
${\bar{\psi}_{\perp}}\cdot\psi_{\perp}={\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}$,
etc.

For (24), $D^{Q}_{\parallel}=qv^{2}n_{\parallel}^{3},\
D^{Q}_{\perp}=qv^{2}n_{\parallel}^{2}n_{\perp}n_{1}$, where
$n_{\parallel}=v\cdot n/|v|,\ n_{\perp}=n_{1}\cdot n$. We also notice

$\displaystyle\begin{split}Q_{abc}v_{c}\bar{\psi}_{a}\psi_{b}&=qn_{a}n_{b}n_{c}v_{a}\psi_{b}\psi_{c}=qvn_{\parallel}^{3}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}+qvn_{\parallel}^{2}n_{\perp}({\bar{\psi}_{\parallel}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\psi_{\parallel}})+qvn_{\parallel}n_{\perp}^{2}{\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}.\end{split}$
(25)

Putting these into (22) and (23), we obtain

$\displaystyle\begin{split}\rho^{\rm signed}_{\rm
spiked}(v,n,q,\beta)=&3^{\frac{N-1}{2}}\pi^{-\frac{N}{2}}\alpha^{\frac{N}{2}}(v^{4}+4\alpha\beta)^{-\frac{1}{2}}(v^{4}+12\alpha\beta)^{-\frac{N-1}{2}}\\\
&\cdot\exp\left[\frac{-\alpha v^{2}+2\alpha qv^{3}n_{\parallel}^{3}-\alpha
q^{2}v^{4}n_{\parallel}^{6}}{v^{4}+4\alpha\beta}-\frac{3\alpha
q^{2}v^{4}n_{\parallel}^{4}n_{\perp}^{2}}{v^{4}+12\alpha\beta}\right]\int
d\bar{\psi}d\psi\,e^{S_{\rm spiked}^{\rm signed}},\end{split}$ (26)

where

$\displaystyle\begin{split}S_{\rm spiked}^{\rm
signed}&=-\left(\frac{v^{4}-4\alpha\beta}{v^{4}+4\alpha\beta}+\frac{8\alpha\beta
qvn_{\parallel}^{3}}{v^{4}+4\alpha\beta}\right){\bar{\psi}_{\parallel}}{\psi_{\parallel}}-\frac{24\alpha\beta
qvn_{\parallel}^{2}n_{\perp}}{v^{4}+12\alpha\beta}\left({\bar{\psi}_{\parallel}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\psi_{\parallel}}\right)+(1-2qvn_{\parallel}n_{\perp}^{2}){\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}\\\
&\ \ +{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}-\frac{8\beta
v^{2}}{v^{4}+12\alpha\beta}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}\left({\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)-\frac{v^{2}}{6\alpha}\left({\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2}.\end{split}$
(27)

It is not difficult to explicitly compute the fermion integration in (26). As
is shown in Appendix B, we obtain

$\displaystyle\begin{split}\int d\bar{\psi}d\psi\,e^{S_{\rm spiked}^{\rm
signed}}=2^{N-6}(-d_{2})^{\frac{N-5}{2}}\bigg{[}&-8d_{2}(-b_{2}^{2}+d_{1}+b_{3}(b_{1}+d_{1})+2b_{1}d_{2})\,U\left(\frac{3-N}{2},\frac{3}{2},-\frac{1}{4d_{2}}\right)\\\
&+2(N-3)(b_{3}d_{1}+2b_{1}d_{2}+6d_{1}d_{2})\,U\left(\frac{5-N}{2},\frac{5}{2},-\frac{1}{4d_{2}}\right)\\\
&-d_{1}(N-3)(N-5)\,U\left(\frac{7-N}{2},\frac{7}{2},-\frac{1}{4d_{2}}\right)\bigg{]},\end{split}$
(28)

where $U$ denotes the confluent hypergeometric function of the second kind,
and $b_{i},d_{i}$ are the coefficients of the terms in (27):

$\displaystyle\begin{split}&b_{1}=-\left(\frac{v^{4}-4\alpha\beta}{v^{4}+4\alpha\beta}+\frac{8\alpha\beta
qvn_{\parallel}^{3}}{v^{4}+4\alpha\beta}\right),\ b_{2}=-\frac{24\alpha\beta
qvn_{\parallel}^{2}n_{\perp}}{v^{4}+12\alpha\beta},\
b_{3}=1-2qvn_{\parallel}n_{\perp}^{2},\\\ &d_{1}=-\frac{8\beta
v^{2}}{v^{4}+12\alpha\beta},\ d_{2}=-\frac{v^{2}}{6\alpha}.\end{split}$ (29)

The result (26) with (28) gives the exact expression of the signed
distribution.

##  4 Distributions

###  4.1 Quantum field theory expression

In this subsection we compute the (authentic) distribution by considering the
determinant factor $|\det M|$ as it is. We take the same procedure as was
employed in [26]. We first introduce bosons and fermions to rewrite $|\det
M|$:

$\displaystyle\begin{split}|\det
M|&=\lim_{\epsilon\rightarrow+0}\frac{\det(M^{2}+\epsilon
I)}{\sqrt{\det(M^{2}+\epsilon I)}}\\\ &=(-\pi)^{-N}\int d\bar{\psi}d\psi
d\bar{\varphi}d\varphi d\phi d\sigma\,e^{-\sigma^{2}-2i\sigma
M\phi-\epsilon\phi^{2}-\bar{\varphi}\varphi-\bar{\psi}M\varphi-\bar{\varphi}M\psi+\epsilon\bar{\psi}\psi},\end{split}$
(30)

where $I$ is an identity matrix of $N$-by-$N$, $\phi_{a},\sigma_{a}$ are real
bosons, $\bar{\psi}_{a},\psi_{a},\bar{\varphi}_{a},\varphi_{a}$ are fermions,
and $\bar{\psi}\psi=\bar{\psi}_{a}\psi_{a}$, etc. Here we have introduced a
positive infinitesimal parameter $\epsilon$ to regularize the expression,
since $M$ may have zero eigenvalues. As in the second line, writing the limit
is suppressed to simplify the notation hereafter, assuming implicitly taking
this limit at ends of computations. In fact the limit turns out to be
straightforward in all the computations of this paper. We have introduced two
sets of bosons and fermions to make the exponent linear in $C$ ($M$ contains
$C$ linearly) for later convenience of the integration over $C$. By performing
similar processes as in Section 3, we obtain

$\displaystyle\rho(v,Q,\beta)=\frac{(-1)^{N}}{2^{N}\pi^{2N}A}\int dCd\lambda
d\bar{\psi}d\psi d\bar{\varphi}d\varphi d\phi d\sigma\,e^{S_{\rm bare}},$ (31)

where

$\displaystyle\begin{split}S_{\rm bare}=&-\alpha
C^{2}-\beta\lambda^{2}+i\lambda_{a}(v_{a}-(C_{abc}+Q_{abc})v_{b}v_{c})\\\
&-\sigma^{2}-2i\sigma_{a}\left(\delta_{ab}-2(Q_{abc}+C_{abc})v_{c}\right)\phi_{b}-\epsilon\phi^{2}\\\
&-\bar{\varphi}\varphi-\bar{\psi}_{a}\left(\delta_{ab}-2(Q_{abc}+C_{abc})v_{c}\right)\varphi_{b}-\bar{\varphi}_{a}\left(\delta_{ab}-2(Q_{abc}+C_{abc})v_{c}\right)\psi_{b}+\epsilon\bar{\psi}\psi.\end{split}$
(32)

As in Section 3, there are no new terms depending on $C$ compared with the
previous case for $Q=\beta=0$ in [26], and therefore the integration over $C$
can be performed as in the previous computation there. Then we obtain a
similar form of the action for $\lambda$ as in Section 3:

$\displaystyle
S_{\lambda}=-\frac{v^{4}}{12\alpha}\lambda_{a}B_{ab}\lambda_{b}+i\lambda_{a}(v_{a}-D_{a}-D_{a}^{Q}),$
(33)

where $B,D^{Q}$ are already defined in (15) and below (13), respectively. Here
$D$ can be taken from [26]999Here $D$ is the sum $D+\tilde{D}$ of [26].:

$\displaystyle\begin{split}D_{a}&=\frac{v^{3}}{3\alpha}\Big{[}({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}})\hat{v}_{a}+\bar{\psi}_{a}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}\varphi_{a}+\bar{\varphi}_{a}{\psi_{\parallel}}+\bar{\varphi_{\parallel}}\psi_{a}+2i\left(\hat{v}_{a}{\sigma_{\parallel}}{\phi_{\parallel}}+\sigma_{a}{\phi_{\parallel}}+{\sigma_{\parallel}}\phi_{a}\right)\Big{]},\end{split}$
(34)

where $\hat{v}_{a}=v_{a}/|v|$. Comparing (33) with (13), the change is to
replace $D^{\rm signed}$ with $-D$. By using (19) with this replacement and
adding the $Q$-dependent but $\lambda$-independent terms in (32), we obtain

$\displaystyle\begin{split}\rho(v,Q,\beta)=&3^{\frac{N-1}{2}}\pi^{-\frac{3N}{2}}\alpha^{\frac{N}{2}}(v^{4}+4\alpha\beta)^{-\frac{1}{2}}(v^{4}+12\alpha\beta)^{-\frac{N-1}{2}}\exp\left[-\frac{\alpha
v^{2}}{v^{4}+4\alpha\beta}\right]\\\ &\cdot\exp\left[\frac{2\alpha
b_{\parallel}vD^{Q}_{\parallel}-\alpha
b_{\parallel}(D^{Q}_{\parallel})^{2}-3\alpha b_{\perp}D^{Q}_{\perp}\cdot
D^{Q}_{\perp}}{v^{4}}\right]Z,\end{split}$ (35)

where $Z$ is a partition function of a quantum field theory,

$\displaystyle Z=(-1)^{N}\int d\bar{\psi}\cdots
d\sigma\,e^{S_{0}+S_{Q,\beta}}.$ (36)

Here $S_{0}$ is the former result in [26] corresponding to $Q=\beta=0$, which
is explicitly given in Appendix C, and

$\displaystyle\begin{split}S_{Q,\beta}=&\frac{2\alpha(b_{\parallel}-1)v-2\alpha
b_{\parallel}D^{Q}_{\parallel}}{v^{4}}D_{\parallel}-\frac{6\alpha
b_{\perp}}{v^{4}}D_{\perp}\cdot
D^{Q}_{\perp}+2Q_{abc}v_{c}\left(\bar{\psi}_{a}\varphi_{b}+\bar{\varphi}_{a}\psi_{b}+2i\sigma_{a}\phi_{b}\right)\\\
&-\frac{\alpha(b_{\parallel}-1)}{v^{4}}D_{\parallel}^{2}-\frac{3\alpha(b_{\perp}-1)}{v^{4}}D_{\perp}\cdot
D_{\perp},\end{split}$ (37)

where $D_{\parallel}=v\cdot D/|v|,\ D_{\perp}=I_{\perp}D$. Note that the first
three terms are some corrections to the kinetic terms, and the latter to the
four-interaction terms. As for $D_{\parallel}$ and $D_{\perp}$, we have more
explicit expressions from (34),

$\displaystyle\begin{split}&D_{\parallel}=\frac{v^{3}}{\alpha}\left({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}+2i{\sigma_{\parallel}}{\phi_{\parallel}}\right),\\\
&D_{\perp}=\frac{v^{3}}{3\alpha}\left({\bar{\psi}_{\perp}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp}}+{\bar{\varphi}_{\perp}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}\psi_{\perp}+2i({\sigma_{\parallel}}{\phi_{\perp}}+{\sigma_{\perp}}{\phi_{\parallel}})\right).\end{split}$
(38)

The four-interaction terms in (37) have the form of self-products. One can
make it quadratic by using the formula
$\frac{1}{\sqrt{\pi}}\int_{\mathbb{R}}dg\,e^{-g^{2}+2Ag}=e^{A^{2}}$. The
result is

$\displaystyle Z=(-1)^{N}\pi^{-\frac{N}{2}}\int
dg_{\parallel}dg_{\perp}d\bar{\psi}\cdots d\sigma\,e^{S_{0}+S_{Q,\beta,g}},$
(39)

where $g_{\parallel}$ is one dimensional, $g_{\perp}$ is $N-1$ dimensional,
and101010Note that $b_{\parallel},b_{\perp}<1$.

$\displaystyle\begin{split}S_{Q,\beta,g}=&-g_{\parallel}^{2}-g_{\perp}^{2}+\left(\frac{2\alpha(b_{\parallel}-1)v-2\alpha
b_{\parallel}D^{Q}_{\parallel}}{v^{4}}+\frac{2\sqrt{\alpha(1-b_{\parallel})}}{v^{2}}g_{\parallel}\right)D_{\parallel}-\frac{6\alpha
b_{\perp}}{v^{4}}D_{\perp}\cdot D^{Q}_{\perp}\\\
&+\frac{2\sqrt{3\alpha(1-b_{\perp})}}{v^{2}}D_{\perp}\cdot
g_{\perp}+2Q_{abc}v_{c}\left(\bar{\psi}_{a}\varphi_{b}+\bar{\varphi}_{a}\psi_{b}+2i\sigma_{a}\phi_{b}\right),\end{split}$
(40)

which contains only quadratic terms of the fields.

###  4.2 Rank-one $Q$

In this subsection we consider the rank-one tensor $Q$ in (24) to explicitly
perform the integration over the fields in (35).

####  4.2.1 A general formula

By putting (24) into (35), one obtains

$\displaystyle\begin{split}\rho(v,Q,\beta)=&3^{\frac{N-1}{2}}\pi^{-\frac{3N}{2}}\alpha^{\frac{N}{2}}(v^{4}+4\alpha\beta)^{-\frac{1}{2}}(v^{4}+12\alpha\beta)^{-\frac{N-1}{2}}\exp\left[-\frac{\alpha
v^{2}}{v^{4}+4\alpha\beta}\right]\\\ &\cdot\exp\left[\frac{2\alpha
qv^{3}n_{\parallel}^{3}-\alpha
q^{2}v^{4}n_{\parallel}^{6}}{v^{4}+4\alpha\beta}-\frac{3\alpha
q^{2}v^{4}n_{\parallel}^{4}n_{\perp}^{2}}{v^{4}+12\alpha\beta}\right]Z,\end{split}$
(41)

where the partition function $Z$ can be computed either by (36) with (37) or
by (39) with (40).

Let us first put (24) into (37). After a lengthy but straightforward
computation using the same decomposition as in Section 3.2, we get

$\displaystyle\begin{split}S_{q,n,\beta}:=&S_{Q=qnnn,\beta}\\\
=&2(qvn_{\parallel}^{3}-1)(1-b_{\parallel})\left({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}+2i{\sigma_{\parallel}}{\phi_{\parallel}}\right)\\\
&+2qvn_{\parallel}^{2}n_{\perp}(1-b_{\perp})\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\perp_{1}}}+2i({\sigma_{\parallel}}{\phi_{\perp_{1}}}+{\sigma_{\perp_{1}}}{\phi_{\parallel}})\right)\\\
&+2qvn_{\parallel}n_{\perp}^{2}\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+2i{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)\\\
&+\frac{8\beta
v^{2}}{v^{4}+4\alpha\beta}\left(-{\bar{\psi}_{\parallel}}{\psi_{\parallel}}{\bar{\varphi}_{\parallel}}{\varphi_{\parallel}}+2i({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}){\sigma_{\parallel}}{\phi_{\parallel}}-2{\sigma_{\parallel}}^{2}{\phi_{\parallel}}^{2}\right)\\\
&+\frac{8\beta
v^{2}}{v^{4}+12\alpha\beta}\bigg{(}{\bar{\psi}_{\perp}}\cdot{\bar{\varphi}_{\perp}}{\psi_{\parallel}}{\varphi_{\parallel}}+\psi_{\perp}\cdot{\varphi_{\perp}}{\bar{\psi}_{\parallel}}{\bar{\varphi}_{\parallel}}-{\bar{\psi}_{\perp}}\cdot\psi_{\perp}{\bar{\varphi}_{\parallel}}{\varphi_{\parallel}}-{\bar{\psi}_{\perp}}\cdot{\varphi_{\perp}}{\bar{\psi}_{\parallel}}{\varphi_{\parallel}}\\\
&\hskip
85.35826pt-{\bar{\varphi}_{\perp}}\cdot{\varphi_{\perp}}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}-{\bar{\varphi}_{\perp}}\cdot\psi_{\perp}{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}\\\
&\hskip
85.35826pt+2i\left({\bar{\psi}_{\perp}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp}}+{\bar{\varphi}_{\perp}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}\psi_{\perp})\cdot({\sigma_{\parallel}}{\phi_{\perp}}+{\sigma_{\perp}}{\phi_{\parallel}}\right)\\\
&\hskip
85.35826pt-2\left({\sigma_{\parallel}}^{2}{\phi_{\perp}}\cdot{\phi_{\perp}}+{\phi_{\parallel}}^{2}{\sigma_{\perp}}\cdot{\sigma_{\perp}}+2{\sigma_{\parallel}}{\phi_{\parallel}}{\phi_{\perp}}\cdot{\sigma_{\perp}}\right)\bigg{)}.\end{split}$
(42)

As for (40), we obtain

$\displaystyle\begin{split}S_{q,n,\beta,g}:=&S_{Q=qnnn,\beta,g}\\\
=&-g_{\parallel}^{2}-g_{\perp}^{2}+2\left((qvn_{\parallel}^{3}-1)(1-b_{\parallel})+v\sqrt{\frac{1-b_{\parallel}}{\alpha}}g_{\parallel}\right)\left({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}+2i{\sigma_{\parallel}}{\phi_{\parallel}}\right)\\\
&+\left(2qvn_{\parallel}^{2}n_{\perp}(1-b_{\perp})\right)\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\perp_{1}}}+2i({\sigma_{\parallel}}{\phi_{\perp_{1}}}+{\sigma_{\perp_{1}}}{\phi_{\parallel}})\right)\\\
&+2qvn_{\parallel}n_{\perp}^{2}\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+2i{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)\\\
&+2v\sqrt{\frac{1-b_{\perp}}{3\alpha}}g_{\perp}\cdot\left({\bar{\psi}_{\perp}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp}}+{\bar{\varphi}_{\perp}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}\psi_{\perp}+2i({\sigma_{\parallel}}{\phi_{\perp}}+{\sigma_{\perp}}{\phi_{\parallel}})\right).\end{split}$
(43)

In the following subsections, we will consider $N=1$, $N=2$ and large-$N$
cases.

####  4.2.2 $N=1$

In this case we ignore all the transverse components, and also set
$n_{\parallel}=1$. By putting these to (35), (39), (43) and (89), and doing
some straightforward computations, we obtain

$\displaystyle\rho(v,q,\beta)=\pi^{-1}\alpha^{\frac{1}{2}}(v^{4}+4\alpha\beta)^{-\frac{1}{2}}\exp\left[\frac{-\alpha
v^{2}+2\alpha qv^{3}-\alpha
q^{2}v^{4}}{v^{4}+4\alpha\beta}\right]\left(\sqrt{\pi}a\,{\rm
Erf}\left(\frac{a}{b}\right)+b\,e^{-\frac{a^{2}}{b^{2}}}\right),$ (44)

where

$\displaystyle\begin{split}&a=1+2(qv-1)(1-b_{\parallel}),\\\
&b=2v\sqrt{\frac{(1-b_{\parallel})}{\alpha}}.\end{split}$ (45)

The details of the derivation are given in Appendix D.

####  4.2.3 $N=2$

In this case the transverse direction is exhausted by one-dimension, namely,
$\perp=\perp_{1}$ and $\perp_{2}$ is null. A special fact about this case is
that the four-interaction terms in (89) have a form of a square:

$\displaystyle
V_{F}+V_{B}+V_{BF}=\frac{v^{2}}{3\alpha}\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+2i{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)^{2}.$
(46)

Therefore we can rewrite this part of the action as

$\displaystyle e^{V_{F}+V_{B}+V_{BF}}=\frac{1}{\sqrt{\pi}}\int
dg\,e^{-g^{2}+2vg\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+2i{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)/\sqrt{3\alpha}},$
(47)

whose exponent contains only quadratic terms of the fields. Using this for
(39), (43) and (89), we obtain

$\displaystyle Z_{N=2}=\pi^{-\frac{3}{2}}\int dg_{1}dg_{2}dg_{3}\int
d\bar{\psi}\cdots
d\sigma\,e^{-g_{1}^{2}-g_{2}^{2}-g_{3}^{2}+K_{\parallel\perp_{1}}},$ (48)

where

$\displaystyle\begin{split}K_{\parallel\perp_{1}}=&-{\bar{\varphi}_{\parallel}}{\varphi_{\parallel}}+\epsilon{\bar{\psi}_{\parallel}}{\psi_{\parallel}}-{\sigma_{\parallel}}^{2}-\epsilon{\phi_{\parallel}}^{2}-{\bar{\varphi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+\epsilon{\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}-{\sigma_{\perp_{1}}}^{2}-\epsilon{\phi_{\perp_{1}}}^{2}\\\
&+a_{1}\left({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}+2i{\sigma_{\parallel}}{\phi_{\parallel}}\right)\\\
&+a_{2}\left({\bar{\psi}_{\parallel}}{\varphi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\parallel}}+2i\left({\sigma_{\parallel}}{\phi_{\perp_{1}}}+{\sigma_{\perp_{1}}}{\phi_{\parallel}}\right)\right)\\\
&+a_{3}\left({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+2i{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)\end{split}$
(49)

with

$\displaystyle\begin{split}&a_{1}=2b_{\parallel}-1+2qv(1-b_{\parallel})n_{\parallel}^{3}+2v\sqrt{\frac{1-b_{\parallel}}{\alpha}}g_{1},\\\
&a_{2}=2qv(1-b_{\perp})n_{\parallel}^{2}n_{\perp}+2v\sqrt{\frac{1-b_{\perp}}{3\alpha}}g_{2},\\\
&a_{3}=-1+2qvn_{\parallel}n_{\perp}^{2}+2v\sqrt{\frac{1}{3\alpha}}g_{3}.\end{split}$
(50)

Then the integration (48) over the fields generates a square root of a
determinant, and we obtain

$\displaystyle Z_{N=2}=\sqrt{\pi}\int
dg_{1}dg_{2}dg_{3}\,e^{-g_{1}^{2}-g_{2}^{2}-g_{3}^{2}}\left|a_{2}^{2}-a_{1}a_{3}\right|.$
(51)

####  4.2.4 Large $N$

For $N>2$ we will not obtain exact expressions of the distributions. We will
rather obtain an expression which is a good approximation for large $N$. For
large $N$ the degrees of freedom carried by the $\perp_{2}$ fields will
dominate over those of the $\parallel\perp_{1}$ fields, since the former is
$(N-2)$-dimensional, while the latter is 2-dimensional. Therefore the dynamics
of the $\perp_{2}$ fields can well be determined by themselves with little
effects from the $\parallel\perp_{1}$ fields, which may be ignored in the
large-$N$ limit. Then the dynamics of the $\parallel\perp_{1}$ fields may be
computed in the backgrounds of the $\perp_{2}$ fields, which can well be
approximated by their classical values because of their large number of
degrees of freedom for large $N$.

More precisely, our approximation is given by

$\displaystyle Z=Z_{\perp_{2}}\,Z_{\parallel\perp_{1}}(R).$ (52)

Here $Z_{\perp_{2}}$ is the partition function determined solely by the
$\perp_{2}$ fields,

$\displaystyle Z_{\perp_{2}}=(-1)^{N-2}\int d{\bar{\psi}_{\perp_{2}}}\cdots
d{\sigma_{\perp_{2}}}\,e^{S_{\perp_{2}}},$ (53)

where $S_{\perp_{2}}$ is the collection of the terms which contain only the
$\perp_{2}$ fields in (88) with (89)111111For instance, we include
$\bar{\psi}_{\perp_{2}}\cdot\psi_{\perp_{2}}\bar{\varphi}_{\perp_{2}}\cdot\varphi_{\perp_{2}}$
but here ignore
$\bar{\psi}_{\perp_{2}}\cdot\psi_{\perp_{2}}\bar{\varphi}_{\perp_{1}}\varphi_{\perp_{1}}$,
$\bar{\psi}_{\perp_{2}}\cdot\psi_{\perp_{2}}\bar{\varphi}_{\parallel}\varphi_{\parallel}$,
etc., because of the reason mentioned in the first paragraph. The ignored
terms will be considered in $Z_{\parallel\perp_{1}}$.. The computation of the
partition function $Z_{\perp_{2}}$ is the same as that in the previous paper
[26], because $S_{\perp_{2}}$ has the same form as the action of the
transverse directions there121212But note the difference of the dimensions of
$\perp_{2}$ here and $\perp$ in [26], where the former is $N-2$, while the
latter is $N-1$. Therefore when we take a result from [26], we have to deduct
$N$ by one..

$Z_{\parallel\perp_{1}}(R)$ is the partition function of the
$\parallel\perp_{1}$ fields in the background of the $\perp_{2}$ fields,

$\displaystyle Z_{\parallel\perp_{1}}(R)=\int d{\bar{\psi}_{\parallel}}\cdots
d\sigma_{\perp_{1}}e^{S_{\parallel\perp_{1}}(R)},$ (54)

where $R$ denotes the classical backgrounds of the $\perp_{2}$ fields, as will
be explained below in more detail. Here the action $S_{\parallel\perp_{1}}(R)$
is composed of all the terms which contain the $\parallel\perp_{1}$ fields in
(42) and (88). Part of the terms in $S_{\parallel\perp_{1}}(R)$ contain the
$\perp_{2}$ fields as well. For large $N$ these $\perp_{2}$ fields may well be
approximated by their classical values because of the large degrees of freedom
of the $\perp_{2}$ fields. For instance, we perform replacements,

$\displaystyle\bar{\psi}_{\perp_{2}}\cdot\varphi_{\perp_{2}}{\bar{\psi}_{\parallel}}{\varphi_{\parallel}}\rightarrow\langle\bar{\psi}_{\perp_{2}}\cdot\varphi_{\perp_{2}}\rangle{\bar{\psi}_{\parallel}}{\varphi_{\parallel}},$
(55)

where $\langle\cdot\rangle$ denotes an expectation value. By doing such
replacements we obtain $S_{\parallel\perp_{1}}(R)$, whose dynamical fields are
only the $\parallel\perp_{1}$ fields.

Obtaining the explicit form of $S_{\parallel\perp_{1}}(R)$ proceeds as
follows. The quadratic and quartic terms of the $\parallel\perp_{1}$ fields
can be processed in the same manner as are performed for $N=2$ in Section
4.2.3, and we obtain (49) with (50). Then the four-interaction terms between
the $\parallel\perp_{1}$ fields and the $\perp_{2}$ fields, where the latter
are replaced by their expectation values like in (55), generate some quadratic
terms of the former, which are explicitly given in (99) of Appendix E. Thus we
have

$\displaystyle
S_{\parallel\perp_{1}}(R)=K_{\parallel\perp_{1}}+V_{\parallel\perp_{1},\perp_{2}}(R),$
(56)

whose terms are all quadratic in the $\parallel\perp_{1}$ fields. Then the
computation of the partition function (54) is just a computation of a
determinant, and we obtain

$\displaystyle Z_{\parallel\perp_{1}}(R)=\sqrt{\pi}\int
dg_{1}dg_{2}dg_{3}\,e^{-g_{1}^{2}-g_{2}^{2}-g_{3}^{2}}\sqrt{\det H},$ (57)

where $H$ is given by

$\displaystyle H=\left(\begin{array}[]{cccc}\epsilon-
A_{1}R_{22}&a_{1}-A_{1}R_{12}&0&a_{2}\\\
a_{1}-A_{1}R_{12}&-1-A_{1}R_{11}&a_{2}&0\\\ 0&a_{2}&\epsilon-
A_{2}R_{22}&a_{3}-A_{2}R_{12}\\\
a_{2}&0&a_{3}-A_{2}R_{12}&-1-A_{2}R_{11}\end{array}\right),$ (62)

where $a_{i}$ are given in (50), $R_{ij}$ are the values of the two point of
correlation functions of the $\perp_{2}$ fields explicitly given in (95) and
(96), and

$\displaystyle A_{1}=\frac{8\beta v^{2}(N-2)}{v^{4}+12\alpha\beta},\
A_{2}=\frac{v^{2}(N-2)}{3\alpha}.$ (63)

The derivation of $H$ is given in Appendix E.

##  5 Comparison with numerical simulations

In this section we compare the distributions obtained for the spiked tensor in
Sections 3 and 4 with Monte Carlo (MC) simulations. The method is basically
the same as that taken in the previous works of the author [24, 25, 26].
Through this section we put $\alpha=1/2$ without loss of generality. In the MC
simulations, all the solutions to the eigenvector equation (1) must be
computed for any randomly sampled $C$ and $\eta$. Since this requires a
reliable polynomial equation solver, we used Mathematica 13 for the MC
simulations. It computes the solutions to (1), which are generally complex,
among which we take only the real ones. To check whether all the solutions are
covered, we checked whether the number of the generally complex solutions to
(1) agreed with the number $2^{N}-1$ of the generally complex eigenvectors
proven in [13], every time the solutions were computed. In fact, when $N$ is
large, we encountered some cases that a few solutions were missing. However,
the missing rates were too small to statistically be relevant for this study.
For example the missing rate was $\lesssim 10^{-4}$ in the $N=9$ data we use
in this paper. We used a workstation which had a Xeon W2295 (3.0GHz, 18
cores), 128GB DDR4 memory, and Ubuntu 20 as OS.

The Monte Carlo simulations were performed by the following procedure.

  * •

Randomly sample $C$ and $\eta$. Each $\eta_{a}$ is randomly sampled by the
normal distribution with the mean value zero and the standard deviation
$\sqrt{2\beta}$. Each $C_{abc}$ is randomly sampled by the normal distribution
with the mean value zero and the standard deviation $1/\sqrt{d_{abc}}$,
corresponding to $\alpha=1/2$, where $d_{abc}$ is the degeneracy factor
defined by131313This degeneracy factor is because the Gaussian term in (5) is
$C_{abc}C_{abc}=\sum_{a\leq b\leq c=1}^{N}d_{abc}C_{abc}^{2}$ in terms of the
independent components of the symmetric tensor $C$.

$\displaystyle d_{abc}=\left\\{\begin{array}[]{ll}1&\hbox{for }a=b=c,\\\
3&\hbox{for }a\neq b=c\hbox{ or }b\neq c=a\hbox{ or }c\neq a=b,\\\ 6&\hbox{for
}a\neq b\neq c\neq a.\end{array}\right.$ (67)

  * •

As explained above, compute all the complex solutions to the eigenvector
equation (1), and pick up only the real ones $v^{i}\ (i=1,2,\cdots,\\#{\rm
sol}(C,\eta))$.

  * •

Store $\left(|v^{i}|,v^{i}\cdot n/|v^{i}|,{\rm sign}\left(\det
M(v^{i},Q,C)\right)\right)$ for $i=1,2,\cdots,\\#{\rm sol}(C,\eta)$.

  * •

Repeat the above processes.

By this sampling procedure, we obtain a series of data,
$\left(|v^{h}|,v^{h}\cdot n/|v^{h}|,{\rm sign}\left(\det
M(v^{h},Q,C)\right)\right)$ for $h=1,2,\cdots,L$, where $L$ denotes the total
number of real solutions obtained141414Note that $L$ is generally different
from $N_{MC}$ below..

To plot the distributions, we classify the data into equally spaced bins in
$v$ and angle $\theta$ as

$\displaystyle\begin{split}&v-\delta_{v}/2<v^{h}\leq v+\delta_{v}/2,\\\
&\cos(\theta-\delta_{\theta}/2)<v^{h}\cdot
n/|v^{h}|\leq\cos(\theta+\delta_{\theta}/2),\\\ &{\rm sign}\left(\det
M(v^{h},Q,C)\right)=1\hbox{ or }{\rm sign}\left(\det
M(v^{h},Q,C)\right)=-1,\end{split}$ (68)

where $v,\theta$ are the center values of a bin, and
$\delta_{v},\delta_{\theta}$ are the sizes of a bin. We denote the total
number of data satisfying (68) as ${\cal
N}_{\delta_{v},\delta_{\theta},+}(v,\theta)$ and ${\cal
N}_{\delta_{v},\delta_{\theta},-}(v,\theta)$ for ${\rm sign}\left(\det
M(v^{h},Q,C)\right)=1$ and ${\rm sign}\left(\det M(v^{h},Q,C)\right)=-1$,
respectively.

Then the distribution of the real eigenvectors from a data is given by

$\displaystyle\rho_{MC}(v,\theta;q,\beta)=\frac{1}{N_{MC}\delta_{v}\delta_{\theta}}\left({\cal
N}_{\delta_{v},\delta_{\theta},+}(v,\theta)+{\cal
N}_{\delta_{v},\delta_{\theta},-}(v,\theta)\pm\sqrt{{\cal
N}_{\delta_{v},\delta_{\theta},+}(v,\theta)+{\cal
N}_{\delta_{v},\delta_{\theta},-}(v,\theta)}\right),$ (69)

where $N_{MC}$ denotes the total number of sampling processes in obtaining the
data and the $\pm$ part represents error estimates. The signed distribution is
given by

$\displaystyle\rho_{MC}^{\rm
signed}(v,\theta;q,\beta)=\frac{1}{N_{MC}\delta_{v}\delta_{\theta}}\left({\cal
N}_{\delta_{v},\delta_{\theta},+}(v,\theta)-{\cal
N}_{\delta_{v},\delta_{\theta},-}(v,\theta)\pm\sqrt{{\cal
N}_{\delta_{v},\delta_{\theta},+}(v,\theta)+{\cal
N}_{\delta_{v},\delta_{\theta},-}(v,\theta)}\right).$ (70)

As for the analytical side, since we take only the size $|v|$ and the relative
angle $\theta$ as data, the above MC distributions should be compared with

$\displaystyle\begin{split}\rho_{\rm
analy}(v,\theta;q,\beta)dvd\theta=&\int_{|v^{\prime}|=v,\,v^{\prime}\cdot
n/|v^{\prime}|=\cos(\theta)}d^{N}v^{\prime}\rho(v^{\prime},q,n,\beta)\\\
=&S_{N-2}v^{N-1}\sin^{N-2}(\theta)\,\rho(v,q,n,\beta)dvd\theta,\end{split}$
(71)

where $S_{N-2}=2\pi^{(N-1)/2}/\Gamma[(N-1)/2]$ is the surface volume of a unit
sphere in the $N-1$-dimensional flat space. Here $\rho(v,q,n,\beta)$ is one of
the expressions obtained in Sections 3 and 4, and $v$ in the argument of
$\rho$ on the righthand side abusively denotes an arbitrary vector
$v^{\prime}$ which satisfies $|v^{\prime}|=v,\,v^{\prime}\cdot
n/|v^{\prime}|=\cos(\theta)$. In the following we will compare the Monte Carlo
and the analytical results.

Figure 1: The MC signed distribution (70) is plotted for a data with
$N=9,\beta=10^{-4},q=10$ and total sampling number $N_{MC}=4\cdot 10^{4}$.

Figure 2: The comparison between the analytical and the MC results with the
same data as of Figure 1. The analytical result is drawn by the solid lines
and the MC results are plotted with error bars. The comparisons are shown for
two example slices in $|v|$ and $\theta$; the left is at $|v|=0.105$ and the
right is at $\theta=\pi/2$.

Let us first consider the signed distribution. The analytical result is
obtained by putting (26) with (28) into (71). Since the analytical result is
an exact result, it should agree with the MC result within errors. In Figure
1, we plot the MC result (70) for $N=9,\beta=10^{-4},q=10$ with $N_{MC}=4\cdot
10^{4}$. As examples, the analytical and MC results are compared at two
slices, one at $|v|=0.105$ and the other at $\theta=\pi/2$ in the two slots of
Figure 2. They agree quite well within error estimates, supporting the
validities of both the analytical and the MC computations.

In Figure 1, an evident negative peak can be observed around $|v|\sim 0.1$ and
$\theta\sim 0.5$. This peak approximately corresponds to an eigenvector
$q^{-1}n_{a}$ of the background tensor $Q_{abc}=q\,n_{a}n_{b}n_{c}$. In fact,
the location satisfies $|v|\sim q^{-1}$, while the angle is not strictly
$\theta=0$. The reason is that the spherical volume is proportional to
$\sin^{N-2}(\theta)$, as in (71), and pushes the peak away from $\theta=0$.
Because of the same reason, the other major structures are concentrated around
$\theta=\pi/2$ in Figure 1. A large-$N$ limit which effectively vanishes this
volume effect will be discussed in Section 6.

In Figure 1 and the right slot of Figure 2 one can also see a peak around
$|v|\sim 0.04,\theta\sim\pi/2$. This peak corresponds to the trivial
eigenvector $v=0$. Because of $\beta>0$ the distribution broadens around
$|v|\sim 0$, and the volume factor $v^{N-1}$ in (71) pushes the peak away from
$|v|=0$.

Figure 3: The MC distribution (69) is plotted for the same data as used in
Figure 1 for $N=9,\beta=10^{-4},q=10$ and $N_{MC}=4\cdot 10^{4}$.

Figure 4: The MC results are plotted with error bars for the same data as in
Figure 3. The analytic result is drawn by the solid lines

In Figure 3 the MC distribution (69) is shown for the same data. Except for
the signs, the characters of the distribution are more or less similar to the
signed case. On the other hand, the analytic result for this case has the
difference that the partition function $Z$ in (35) is computed by the
approximation (52), while it was exact for the signed case. The exact
expression of $Z_{\perp_{2}}$ can be taken from the previous result in [26],
which is explicitly given in Appendix F. As for $Z_{\parallel\perp_{1}}$, by
numerically integrating (57) on a grid of points in $|v|$ and $\theta$, an
interpolation function of $Z_{\parallel\perp_{1}}$ is computed and used. In
Figure 4 the analytic and the MC results are compared. The agreement is fairly
satisfactory except for some slight systematic deviations around a peak.

##  6 Large-$N$ limit

In this section we will take large-$N$ limits of the distribution obtained in
Section 4.2 for a spiked tensor. We will particularly pay attention to the
parameter region where the peak corresponding to the background $Q$ can been
seen in the eigenvector distribution. We will consider two large-$N$ limits.
In one large-$N$ limit, we will derive the result that a peak can be well
identified with $Q$ for the parameter region, $\alpha q^{2}/N\gtrsim 0.6,\beta
q^{2}N\lesssim 0.1$. In particular for $\beta q^{2}=0$, we will find the
threshold value to be $0.66<(\alpha q^{2}/N)_{c}<0.67$, which agrees with
Proposition 2 of [29]. However this peak is always smaller than the other
peak(s) at ${n_{\parallel}}=0$ and therefore relatively vanishes in the strict
large-$N$ limit. In the other scaling limit, $\alpha q^{2}\sim
N^{\gamma},\beta q^{2}\sim N^{-\gamma}$ with $\gamma>1$, the peak remains in
the strict large-$N$ limit.

We will consider large-$N$ limits which keep both the parameters $Q$ and
$\beta$ relevant. As was discussed in Section 5, the volume factor
$\sin^{N-2}\theta$ in (71) suppresses the peak of the eigenvector $q^{-1}n$ of
the background tensor $Q$, and this suppression becomes stronger as $N$
becomes larger. Therefore, to obtain an interesting large-$N$ limit, the
parameters must be scaled so as to compete with $\sin^{N-2}\theta\sim
e^{N\log(\sin\theta)}$. A large-$N$ scaling which makes the exponential factor
in (41) in this order is given by

$\displaystyle\alpha=\frac{N\tilde{\alpha}}{q^{2}},\
\beta=\frac{\tilde{\beta}}{Nq^{2}},\ v=\frac{\tilde{v}}{q},$ (72)

where ${\tilde{\alpha}},{\tilde{\beta}}$ are kept finite. Here the factors of
$q$ are to absorb the dependence on $q$ from the formulas below.

Let us discuss the large-$N$ limit of $Z=Z_{\perp_{2}}Z_{\parallel\perp_{1}}$
in Section 4.2.4. The large-$N$ limit of $Z_{\perp_{2}}$ was determined in
[25], and it is given by

$\displaystyle Z^{N=\infty}_{\perp_{2}}\sim{\rm
const.}\,e^{NS^{\infty}_{\perp_{2}}},$ (73)

where151515For simplicity, $S^{\infty}_{\perp_{2}}$ is shifted by an
irrelevant constant from the corresponding expression with $R=1/2$ in [25].

$\displaystyle S^{\infty}_{\perp_{2}}(x)=\left\\{\begin{array}[]{ll}\log
2+\log(x)+\frac{1-\sqrt{1-4x}}{4x}-\log\left(1-\sqrt{1-4x}\right)&\hbox{ for
}0<x<\frac{1}{4},\\\ \frac{1}{4x}+\frac{1}{2}\log(x)&\hbox{ for
}\frac{1}{4}<x,\end{array}\right.$ (76)

with161616$N$ must be deducted by one, when we take a result from [25]. See a
footnote below (94).
$x=(N-2)v^{2}/(3\alpha)\sim\tilde{v}^{2}/(3\tilde{\alpha})$. As for
$Z_{\parallel\perp_{1}}$, one can easily see that the limit of (57) is just
given by dropping the terms dependent on $g_{i}$ in (50), while the
$N$-dependencies of $A_{i}$ in (63) and $R_{ij}$ in (95) and (96) drop out.
Therefore $H$ does not depend on $g_{i}$ and we get

$\displaystyle Z_{\parallel\perp_{1}}^{N=\infty}=\pi^{2}\left.\sqrt{\det
H}\right|_{g_{i}=0},$ (77)

which does not have a relevant effect to the formula below for large-$N$.

By collecting the results above and using (71) and (41), we obtain

$\displaystyle\begin{split}S_{\infty}(\tilde{v},\theta)=&\lim_{N\rightarrow\infty}\frac{1}{N}\log\rho_{\rm
analy}\\\
=&\hbox{const.}+S_{\perp_{2}}^{\infty}+\log{\tilde{v}}+\log({n_{\perp}})-\frac{1}{2}\log({\tilde{v}}^{4}+12{\tilde{\alpha}}{\tilde{\beta}})\\\
&+\frac{-{\tilde{\alpha}}{\tilde{v}}^{2}+2{\tilde{\alpha}}{\tilde{v}}^{3}{n_{\parallel}}^{3}-{\tilde{\alpha}}{\tilde{v}}^{4}{n_{\parallel}}^{6}}{{\tilde{v}}^{4}+4{\tilde{\alpha}}{\tilde{\beta}}}-\frac{3{\tilde{\alpha}}{\tilde{v}}^{4}{n_{\parallel}}^{4}{n_{\perp}}^{2}}{{\tilde{v}}^{4}+12{\tilde{\alpha}}{\tilde{\beta}}},\end{split}$
(78)

where $n_{\parallel}=\cos\theta$ ($n_{\perp}=\sin\theta$), and const. is the
part not dependent on ${\tilde{v}}$ or $\theta$.

Figure 5: In the shaded region of the parameters, the eigenvector distribution
has a peak of $S_{\infty}$ corresponding to the eigenvector of $Q$.

Figure 6: In the left slot, $S_{\infty}$ (const. being ignored) is plotted for
$\log_{10}{\tilde{\alpha}}=-1,\log_{10}{\tilde{\beta}}=1$, which is in the
unshaded region of Figure 5. The right slot is for
$\log_{10}{\tilde{\alpha}}=1,\log_{10}{\tilde{\beta}}=-2$ in the shaded
region. In the latter case, a peak near ${\tilde{v}}\sim 1,\theta\sim 0$
corresponding to the eigenvector of $Q$ can be found. The tiny gaps in the
plots are not essential; They seem to be caused by the drawing program
(Mathematica) avoiding the singularity at $x=1/4$ in (76), where the function
is continuous but its first derivative is discrete.

Figure 7: The values of ${n_{\parallel}}$(left) and ${\tilde{v}}$(right) of
the peak with $n_{\parallel}>0$, corresponding to the eigenvector of $Q$, are
plotted. Identification of this peak with $Q$ can well be done in the region
$\log_{10}\tilde{\alpha}\gtrsim-0.2$ and $\log_{10}\tilde{\beta}\lesssim-1$.

It is interesting to study the profile of $S_{\infty}({\tilde{v}},\theta)$ in
the ${\tilde{v}}$ and $\theta$ plane for various values of
$\tilde{\alpha},\tilde{\beta}$. We have numerically studied it for the
parameter region $10^{-3}\leq\tilde{\alpha}\leq
10^{3},10^{-3}\leq\tilde{\beta}\leq 10^{3}$. In the unshaded region of Figure
5, the peak(s) exist only along $n_{\parallel}=0$, as is shown in the left
slot of Figure 6 as an example. In the shaded region, in addition to the
peak(s) at $n_{\parallel}=0$, there exists also a peak which has non-zero
$n_{\parallel}$. This peak corresponds to the eigenvector $q^{-1}n$ of the
background tensor $Q$, as is shown in the right slot of Figure 6. In Figure 7,
the values of $n_{\parallel}$ and $\tilde{v}$ are plotted for the latter peak.
The location can be well identified with $q^{-1}n$, if the values take
$n_{\parallel}\sim 1$ and $\tilde{v}\sim 1$. As can be seen in the plots, this
occurs in the region, $\log_{10}\tilde{\alpha}\gtrsim-0.2$ and
$\log_{10}\tilde{\beta}\lesssim-1$. This is the parameter region in which the
background tensor $Q$ can be detected well.

Figure 8: The values of ${n_{\parallel}}$ and ${\tilde{v}}$ of the peak with
${n_{\parallel}}>0$ for ${\tilde{\beta}}=0$. The threshold value is
$0.66<{\tilde{\alpha}}_{c}<0.67$.

It is interesting to compare this detectable region with a result of [29]. As
can be seen in Figure 5, the shaded region has an edge around
$\log_{10}{\tilde{\alpha}}\sim-0.2$, namely, ${\tilde{\alpha}}\sim 0.63$,
independent of ${\tilde{\beta}}$ for $\log_{10}{\tilde{\beta}}\lesssim-1$. To
see the threshold value more precisely for ${\tilde{\beta}}=0$, we plot
$n_{\parallel}$ and ${\tilde{v}}$ of the peak with $n_{\parallel}>0$ in Figure
8. We find that the peak does not exist at $\tilde{\alpha}\leq 0.66$, but
exists at $\tilde{\alpha}\geq 0.67$ with ${n_{\parallel}}\gtrsim 0.7$. On the
other hand, as explained in Appendix G, Proposition 2 of [29] states that the
threshold value is ${\tilde{\alpha}}=2/3$, which indeed agrees with our value.

We numerically observed that a peak at ${n_{\parallel}}=0$ always take the
largest value of $S_{\infty}$ at least in the parameter region of
$\tilde{\alpha},\tilde{\beta}$ we have mentioned above. This means that,
because $\rho\sim e^{NS_{\infty}}$, the peak corresponding to $Q$ will
effectively be invisible compared to the peak(s) at ${n_{\parallel}}=0$ in the
strict large-$N$ limit. Therefore in the strict large-$N$ limit, $Q$, namely a
“signal”, cannot be detected by solving the eigenvector equation (1).

Figure 9: $S_{\infty}^{\gamma}$ is plotted for $\log_{10}\tilde{\alpha}=0$ and
$\log_{10}\tilde{\beta}=-1$ as an example. There is a peak corresponding to
the eigenvector of the background tensor $Q$.

The main reason for the above difficulty of detection comes from the strong
effect of the volume factor $\sin^{N-2}\theta$ in (71), which enhances the
region ${n_{\parallel}}\sim 0$ so strongly. Therefore an obvious way to solve
this difficulty is to consider another scaling limit which overwhelms the
volume factor. An example is given by

$\displaystyle\alpha=\frac{N^{\gamma}\tilde{\alpha}}{q^{2}},\
\beta=\frac{\tilde{\beta}}{N^{\gamma}q^{2}},\ v=\frac{\tilde{v}}{q},\
\gamma>1.$ (79)

In this limit, $x=(N-2)v^{2}/(3\alpha)\sim N^{-p+1}\rightarrow 0$ in the
large-$N$ limit, so therefore (76) becomes a constant, meaning that
$Z_{\perp_{2}}$ is a free theory independent of $v$. As for
$Z_{\parallel\perp_{1}}$, $A_{i}\rightarrow 0$ and $R_{ij}$ approaches finite
values, so $Z_{\parallel\perp_{1}}$ is again a finite quantity. Therefore from
(41), the major contribution comes only from the exponent, and we obtain

$\displaystyle\begin{split}S_{\infty}^{\gamma}(\tilde{v},\theta)=&\lim_{N\rightarrow\infty}\frac{1}{N^{\gamma}}\log\rho_{\rm
analy}\\\
=&\frac{-{\tilde{\alpha}}{\tilde{v}}^{2}+2{\tilde{\alpha}}{\tilde{v}}^{3}{n_{\parallel}}^{3}-{\tilde{\alpha}}{\tilde{v}}^{4}{n_{\parallel}}^{6}}{{\tilde{v}}^{4}+4{\tilde{\alpha}}{\tilde{\beta}}}-\frac{3{\tilde{\alpha}}{\tilde{v}}^{4}{n_{\parallel}}^{4}{n_{\perp}}^{2}}{{\tilde{v}}^{4}+12{\tilde{\alpha}}{\tilde{\beta}}}.\end{split}$
(80)

As is shown in Appendix H, it is straightforward to prove that the maximum
value of $S_{\infty}^{\gamma}$ is $0$, and this occurs only at three
locations: (i) $\tilde{v}=0$, (ii)
$\tilde{v}\rightarrow\infty,n_{\parallel}=0$, (iii)
$\tilde{v}=1,n_{\parallel}=1$. The last location corresponds to the background
$Q$. An example of $S^{\gamma}_{\infty}$ is shown in Figure 9. Since the
eigenvector distribution is given by $\rho\sim
e^{N^{\gamma}S_{\infty}^{\gamma}}$, there remains only the three locations
above in the strict large-$N$ limit. This means that, in the limit, a finite
eigenvector ($v\neq 0,\infty$) is surely that of the background $Q$.

##  7 Summary and future prospects

In this paper we have studied the real eigenvector distributions of real
symmetric order-three Gaussian random tensors in the case that the random
tensors have non-zero mean value backgrounds and the eigenvector equations
have Gaussian random deviations. This is an extension of the previous studies
[24, 25, 26], which have no such mean values or deviations. We have derived
the quantum field theories with quartic interactions whose partition functions
give the distributions. For the background tensor being rank-one (a spiked
tensor case) in particular, we have explicitly derived the distributions by
computing the partition functions exactly or approximately. We have obtained
good agreement between the analytical results and Monte Carlo simulations. We
have derived the scaling and range of parameters for the background tensor to
be detectable in the distributions in the large-$N$ limit. Our threshold value
has agreed with that of [29].

The quantum field theories we have derived in this paper are much more
complicated than those in the previous studies [24, 25, 26] due to the
presence of the backgrounds and the deviations. Nonetheless, we have obtained
some exact expressions for the signed distributions, and have also derived
some approximate expressions of the (authentic) distributions, which agree
very well with the Monte Carlo results. This success can be ascribed to the
quantum field theoretical expressions, to which we can apply various well-
developed techniques and knowledge of quantum field theories. The results of
this paper strengthen our belief that the quantum field theoretical procedure
for computing distributions of quantities in random tensors is general,
powerful, and intuitive.

As far as random tensors are Gaussian, it is in principle straightforward to
extend the quantum field theoretical procedure to some other problems in
random tensors; distributions of complex eigenvectors/values, tensor rank
decompositions, correlations among eigenvectors, etc. Although derived quantum
field theories with quartic interactions may become quite complicated, it will
always be possible to find ways to compute the partition functions by quantum
field theoretical techniques, knowledge and intuition. These studies will
enrich fundamental knowledge about random tensors, which will eventually be
applied in various subjects in future studies.

Tensor models have emerged from discrete approaches to quantum gravity, and
are also taking active part in more recent approaches, such as in the AdS/CFT
correspondence [33]. A question of the author’s interest is whether there
exists an analogous phenomenon in random tensors as the Gross-Witten-Wadia
transition [4, 5]. In fact there are some indications that similar transitions
exist in the context of a discrete model of quantum gravity [34, 35]. We hope
the knowledge about random tensors enriched along the line of our studies will
give some insights into quantum gravity in the future.

## Acknowledgements

The author is supported in part by JSPS KAKENHI Grant No.19K03825. He would
like to thank O. Evnin and N. Delprote for some stimulating discussions.

##  Appendix Appendix A Derivation of (22)

From (14), the parallel/transverse parts of $D^{\rm signed}$ are given by

$\displaystyle\begin{split}D^{\rm
signed}_{\parallel}&=\frac{v^{3}}{\alpha}{\bar{\psi}_{\parallel}}{\psi_{\parallel}},\\\
D^{\rm
signed}_{\perp}&=\frac{v^{3}}{3\alpha}\left({\bar{\psi}_{\perp}}{\psi_{\parallel}}+{\bar{\psi}_{\parallel}}\psi_{\perp}\right).\end{split}$
(81)

By putting (81) into (19), we obtain

$\displaystyle\begin{split}\delta S^{\rm signed}_{\lambda}-\delta S^{\rm
signed}_{\lambda}(Q=\beta=0)&=\frac{1}{2}\log b_{\parallel}+\frac{N-1}{2}\log
b_{\perp}-\frac{\alpha(b_{\parallel}-1)}{v^{2}}+\frac{2\alpha
b_{\parallel}D^{Q}_{\parallel}}{v^{3}}-\frac{\alpha
b_{\parallel}(D^{Q}_{\parallel})^{2}}{v^{4}}\\\ &\ \ -\frac{3\alpha
b_{\perp}D^{Q}_{\perp}\cdot
D^{Q}_{\perp}}{v^{4}}-2\left(b_{\parallel}-1-\frac{b_{\parallel}D^{Q}_{\parallel}}{v}\right){\bar{\psi}_{\parallel}}{\psi_{\parallel}}\\\
&\ \
+\frac{2b_{\perp}}{v}D^{Q}_{\perp}\cdot\left({\bar{\psi}_{\perp}}{\psi_{\parallel}}+{\bar{\psi}_{\parallel}}\psi_{\perp}\right)+\frac{2v^{2}(b_{\perp}-1)}{3\alpha}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}{\bar{\psi}_{\perp}}\cdot\psi_{\perp}.\end{split}$
(82)

Adding this and the last term of (12) to (21), one obtains (22) with (23).

##  Appendix Appendix B Derivation of (28)

Let us parametrize (27) as follows:

$\displaystyle\begin{split}S_{b,d,k}&=b_{1}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}+b_{2}\left({\bar{\psi}_{\parallel}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\psi_{\parallel}}\right)+b_{3}{\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+k{\bar{\psi}_{\perp_{2}}}{\psi_{\perp_{2}}}\\\
&\ \
+d_{1}\left({\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{2}}}{\psi_{\perp_{2}}}\right){\bar{\psi}_{\parallel}}{\psi_{\parallel}}+d_{2}\left(\bar{\psi}_{1}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{2}}}{\psi_{\perp_{2}}}\right)^{2}.\end{split}$
(83)

Then by explicitly performing the fermion integrations for $\parallel$ and
$\perp_{1}$ directions, we obtain

$\displaystyle\begin{split}&\int d\bar{\psi}d\psi\,e^{S_{b,d,k}}\\\ &=\int
d{\bar{\psi}_{\perp_{2}}}d{\psi_{\perp_{2}}}\left(d_{1}+b_{1}b_{3}-b_{2}^{2}+(2b_{1}d_{2}+b_{3}d_{1}){\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}+2d_{1}d_{2}\left({\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2}\right)e^{k{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}+d_{2}\left({\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2}}\\\
&=\left(d_{1}+b_{1}b_{3}-b_{2}^{2}+(2b_{1}d_{2}+b_{3}d_{1})\frac{\partial}{\partial
k}+2d_{1}d_{2}\frac{\partial^{2}}{\partial k^{2}}\right)\int
d{\bar{\psi}_{\perp_{2}}}d{\psi_{\perp_{2}}}\,e^{k{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}+d_{2}\left({\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2}}.\end{split}$
(84)

Now the last fermion integration can be computed as

$\displaystyle\begin{split}\int
d{\bar{\psi}_{\perp_{2}}}d{\psi_{\perp_{2}}}\,e^{k{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}+d_{2}\left({\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2}}&=\sum_{n=0}^{\infty}\frac{d_{2}^{n}}{n!}\left({\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\right)^{2n}\int
d{\bar{\psi}_{\perp_{2}}}d{\psi_{\perp_{2}}}\,e^{k{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}}\\\
&=\sum_{n=0}^{\lfloor\frac{N-2}{2}\rfloor}\frac{d_{2}^{n}}{n!}\frac{\partial^{2n}}{\partial
k^{2n}}k^{N-2}\\\
&=(-4d_{2})^{\frac{N-3}{2}}k\,U\left(\frac{3-N}{2},\frac{3}{2},-\frac{k^{2}}{4d_{2}}\right),\end{split}$
(85)

where $U$ is the confluent hypergeometric function of the second kind. The
last equality can be shown by using the following relation to a hypergeometric
function and comparing with its asymptotic expansion:

$\displaystyle U(a,b,z)\sim z^{-a}\,{}_{2}F_{0}(a,1+a-b,-z^{-1}),$ (86)

where the hypergeometric function has a formal series expansion,

$\displaystyle{}_{2}F_{0}(a,b,x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{n!}x^{n}$
(87)

with the Pochhammer symbol, $(a)_{n}=a(a+1)\cdots(a+n-1)$ ($(a)_{0}=1$). For
the argument in (85), the formal series stops at finite $n$, and hence (86) is
an exact relation. One can also find the confluent hypergeometric function can
be expressed by an hermite polynomial.

##  Appendix Appendix C Explicit form of $S_{0}$

The $Q=\beta=0$ case was studied in [26], and the action for this case is
given by

$\displaystyle S_{0}=K_{F}+K_{B}+V_{F}+V_{B}+V_{BF},$ (88)

where

$\displaystyle\begin{split}&K_{F}=-{\bar{\varphi}_{\perp}}\cdot{\varphi_{\perp}}-{\bar{\psi}_{\perp}}\cdot{\varphi_{\perp}}-{\bar{\varphi}_{\perp}}\cdot\psi_{\perp}+\epsilon{\bar{\psi}_{\perp}}\cdot\psi_{\perp}-{\bar{\varphi}_{\parallel}}\cdot{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}\cdot{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}\cdot{\psi_{\parallel}}+\epsilon{\bar{\psi}_{\parallel}}\cdot{\psi_{\parallel}},\\\
&K_{B}=-{\sigma_{\perp}}^{2}-2i{\sigma_{\perp}}\cdot{\phi_{\perp}}-\epsilon{\phi_{\perp}}^{2}-{\sigma_{\parallel}}^{2}+2i{\sigma_{\parallel}}\cdot{\phi_{\parallel}}-\epsilon{\phi_{\parallel}}^{2},\\\
&V_{F}=-\frac{v^{2}}{6\alpha}\left(({\bar{\psi}_{\perp}}\cdot{\varphi_{\perp}})^{2}+({\bar{\varphi}_{\perp}}\cdot\psi_{\perp})^{2}+2{\bar{\psi}_{\perp}}\cdot{\bar{\varphi}_{\perp}}{\varphi_{\perp}}\cdot\psi_{\perp}+2{\bar{\psi}_{\perp}}\cdot\psi_{\perp}{\bar{\varphi}_{\perp}}\cdot{\varphi_{\perp}}\right),\\\
&V_{B}=-\frac{2v^{2}}{3\alpha}\left({\sigma_{\perp}}^{2}{\phi_{\perp}}^{2}+({\sigma_{\perp}}\cdot{\phi_{\perp}})^{2}\right),\\\
&V_{BF}=\frac{2iv^{2}}{3\alpha}\left({\bar{\psi}_{\perp}}\cdot{\sigma_{\perp}}{\varphi_{\perp}}\cdot{\phi_{\perp}}+{\bar{\varphi}_{\perp}}\cdot{\sigma_{\perp}}\psi_{\perp}\cdot{\phi_{\perp}}+{\bar{\psi}_{\perp}}\cdot{\phi_{\perp}}{\varphi_{\perp}}\cdot{\sigma_{\perp}}+{\bar{\varphi}_{\perp}}\cdot{\phi_{\perp}}\psi_{\perp}\cdot{\sigma_{\perp}}\right).\end{split}$
(89)

Note that the kinetic terms of the parallel and the transverse components of
the fields respectively have slightly different sign structures, and that the
four-interactions exist only among the transverse components.

##  Appendix Appendix D Distribution for $N=1$ rank-one $Q$

In this appendix, we derive (44). Ignoring all the transverse components, and
setting $n_{\parallel}=1$ in (35), (39), (43), and (89), we obtain

$\displaystyle\rho(v,q,\beta)=\pi^{-2}\alpha^{\frac{1}{2}}(v^{4}+12\alpha\beta)^{-\frac{1}{2}}\exp\left[\frac{-\alpha
v^{2}+2\alpha qv^{3}-\alpha q^{2}v^{4}}{v^{4}+4\alpha\beta}\right](-1)\int
dgd\bar{\psi}\cdots d\sigma\,e^{S_{N=1}},$ (90)

where

$\displaystyle\begin{split}S_{N=1}=-g^{2}-{\bar{\varphi}_{\parallel}}\cdot{\varphi_{\parallel}}+(a+b\,g)({\bar{\psi}_{\parallel}}\cdot{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}\cdot{\psi_{\parallel}})+\epsilon{\bar{\psi}_{\parallel}}\cdot{\psi_{\parallel}}-{\sigma_{\parallel}}^{2}+2i(a+b\,g){\sigma_{\parallel}}\cdot{\phi_{\parallel}}-\epsilon{\phi_{\parallel}}^{2}\end{split}$
(91)

with $a,b$ given in (45).

The boson-fermion integration in (90) produces a square root of the
determinant of a two-by-two matrix. It is easy to see that the
$\epsilon\rightarrow+0$ limit is smooth, and we obtain

$\displaystyle\begin{split}(-1)\int dgd\bar{\psi}\cdots
d\sigma\,e^{S_{N=1}}&=\pi\int dg\,e^{-g^{2}}\left|a+b\,g\right|\\\
&=\pi\left(\sqrt{\pi}a\,{\rm
Erf}\left(\frac{a}{b}\right)+b\,e^{-\frac{a^{2}}{b^{2}}}\right)\end{split}$
(92)

with the error function Erf.

##  Appendix Appendix E Interactions between the $\parallel\perp_{1}$ and
$\perp_{2}$ fields

There are no quadratic terms containing one $\parallel\perp_{1}$ field and one
$\perp_{2}$ field, because the index of the $\perp_{2}$ field cannot be
contracted with $v$ or $n$. Therefore the $\parallel\perp_{1}$ fields can
couple with the $\perp_{2}$ fields only through the four-interaction terms in
(42) and (89). By noting that $X_{\perp}\cdot
Y_{\perp}=X_{\perp_{1}}Y_{\perp_{1}}+X_{\perp_{2}}\cdot Y_{\perp_{2}}$ for
arbitrary fields $X,Y$, and collecting all the interaction terms between the
$\parallel\perp_{1}$ and the $\perp_{2}$ fields, we obtain

$\displaystyle\begin{split}V_{\parallel\perp_{1},\perp_{2}}=&\frac{8\beta
v^{2}}{v^{4}+12\alpha\beta}\big{(}{\bar{\psi}_{\perp_{2}}}\cdot{\bar{\varphi}_{\perp_{2}}}{\psi_{\parallel}}{\varphi_{\parallel}}+{\psi_{\perp_{2}}}\cdot{\varphi_{\perp_{2}}}{\bar{\psi}_{\parallel}}{\bar{\varphi}_{\parallel}}-{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}{\bar{\varphi}_{\parallel}}{\varphi_{\parallel}}\\\
&\hskip
56.9055pt-{\bar{\psi}_{\perp_{2}}}\cdot{\varphi_{\perp_{2}}}{\bar{\psi}_{\parallel}}{\varphi_{\parallel}}-{\bar{\varphi}_{\perp_{2}}}\cdot{\varphi_{\perp_{2}}}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}-{\bar{\varphi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}{\bar{\varphi}_{\parallel}}{\psi_{\parallel}}\big{)}\\\
&+\frac{16\beta
v^{2}i}{v^{4}+12\alpha\beta}\left({\bar{\psi}_{\perp_{2}}}{\varphi_{\parallel}}+{\bar{\psi}_{\parallel}}{\varphi_{\perp_{2}}}+{\bar{\varphi}_{\perp_{2}}}{\psi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\perp_{2}}}\right)\left({\sigma_{\parallel}}{\phi_{\perp_{2}}}+{\sigma_{\perp_{2}}}{\phi_{\parallel}}\right)\\\
&-\frac{16\beta
v^{2}}{v^{4}+12\alpha\beta}\left({\sigma_{\parallel}}^{2}{\phi_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}+{\phi_{\parallel}}^{2}{\sigma_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}+2{\sigma_{\parallel}}{\phi_{\parallel}}{\phi_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}\right)\\\
&-\frac{v^{2}}{3\alpha}\big{(}{\bar{\psi}_{\perp_{2}}}\cdot{\varphi_{\perp_{2}}}{\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\bar{\varphi}_{\perp_{1}}}{\varphi_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}\\\
&\hskip
28.45274pt+{\bar{\psi}_{\perp_{2}}}\cdot{\bar{\varphi}_{\perp_{2}}}{\varphi_{\perp_{1}}}{\psi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}{\bar{\varphi}_{\perp_{2}}}\cdot{\varphi_{\perp_{2}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\psi_{\perp_{2}}}{\bar{\varphi}_{\perp_{1}}}{\varphi_{\perp_{1}}}\big{)}\\\
&-\frac{2v^{2}}{3\alpha}\left({\sigma_{\perp_{1}}}^{2}{\phi_{\perp_{2}}}^{2}+{\sigma_{\perp_{2}}}^{2}{\phi_{\perp_{1}}}^{2}+2{\sigma_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}{\sigma_{\perp_{1}}}{\phi_{\perp_{1}}}\right)\\\
&+\frac{2v^{2}i}{3\alpha}\big{(}{\bar{\psi}_{\perp_{1}}}{\sigma_{\perp_{1}}}{\varphi_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}{\varphi_{\perp_{1}}}{\phi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\sigma_{\perp_{1}}}{\psi_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}\\\
&\hskip
42.67912pt+{\bar{\varphi}_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}{\psi_{\perp_{1}}}{\phi_{\perp_{1}}}+{\bar{\psi}_{\perp_{1}}}{\phi_{\perp_{1}}}{\varphi_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}+{\bar{\psi}_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}{\varphi_{\perp_{1}}}{\sigma_{\perp_{1}}}\\\
&\hskip
42.67912pt+{\bar{\varphi}_{\perp_{1}}}{\phi_{\perp_{1}}}{\psi_{\perp_{2}}}\cdot{\sigma_{\perp_{2}}}+{\bar{\varphi}_{\perp_{2}}}\cdot{\phi_{\perp_{2}}}{\psi_{\perp_{1}}}{\sigma_{\perp_{1}}}\big{)}.\end{split}$
(93)

The expectation values of the $\perp_{2}$ fields can be taken from the
large-$N$ Schwinger-Dyson analysis performed in [25]. The results were171717To
avoid duplication of notations, we use $R_{ij}$ in place of $Q_{ij}$ of [25].
Another thing to note is that, though we have both bosons and fermions in the
current system, which is different from the setup of [25], the leading-order
Schwinger-Dyson analysis of the current system turns out to lead to the same
two-fermion expectation values as [25]. The reason is the presence of the
supersymmetry shown below, which assures the two-boson expectation values are
just the copies of those of fermions.

$\displaystyle\begin{split}&\langle{\bar{\psi}_{\perp_{2}}}{}_{a}{\psi_{\perp_{2}}}{}_{b}\rangle=R_{11}\delta_{ab},\\\
&\langle{\bar{\psi}_{\perp_{2}}}{}_{a}{\varphi_{\perp_{2}}}{}_{b}\rangle=R_{12}\delta_{ab},\\\
&\langle{\bar{\varphi}_{\perp_{2}}}{}_{a}{\psi_{\perp_{2}}}{}_{b}\rangle=R_{21}\delta_{ab},\\\
&\langle{\bar{\varphi}_{\perp_{2}}}{}_{a}{\varphi_{\perp_{2}}}{}_{b}\rangle=R_{22}\delta_{ab},\\\
&\hbox{Others}=0,\end{split}$ (94)

where, with a newly introduced parameter181818As the dimension of $\perp_{2}$
is $N-2$, the formula presented in [25] for the dimension $N-1$ of $\perp$
must be replaced with $N-2$. $x=v^{2}(N-2)/(3\alpha)$,

  * •

$0<x<1/4$

$\displaystyle\begin{split}&R_{11}=\frac{-\sqrt{1-4x}+1}{2x\sqrt{1-4x}},\\\
&R_{12}=R_{21}=\frac{1-\sqrt{1-4x}-4x}{2x\sqrt{1-4x}},\\\
&R_{22}=0,\end{split}$ (95)

  * •

$x>1/4$

$\displaystyle\begin{split}&R_{11}=\frac{\sqrt{-1+4x}}{2\sqrt{\epsilon}x}-\frac{1}{2x}+{\cal
O}(\sqrt{\epsilon}),\\\
&R_{12}=R_{21}=-\frac{1}{2x}+\frac{\sqrt{\epsilon}}{2x\sqrt{-1+4x}}+{\cal
O}(\epsilon^{\frac{3}{2}}),\\\
&R_{22}=-\frac{\sqrt{\epsilon}\sqrt{-1+4x}}{2x}+\frac{\epsilon}{2x}+{\cal
O}(\epsilon^{\frac{3}{2}}).\end{split}$ (96)

The two-boson expectation values can also be represented by $R_{ij}$ by
assuming that a supersymmetry is not spontaneously broken. It is easy to check
that $S_{0}+S_{Q,\beta}$ from (88) and (37) are invariant under the following
supersymmetry transformation:

$\displaystyle\delta\psi_{a}=-\phi_{a},\
\delta\phi_{a}=\frac{1}{2}\bar{\psi}_{a},\ \delta\varphi_{a}=i\sigma_{a},\
\delta\sigma_{a}=\frac{i}{2}\bar{\varphi}_{a},\ \delta\hbox{(others)}=0.$ (97)

By assuming the non-breaking of the supersymmetry, we obtain for instance a
relation,
$0=\langle\delta(\psi_{a}\sigma_{b})\rangle=-\langle\phi_{a}\sigma_{b}\rangle-\langle\psi_{a}\frac{i}{2}\bar{\varphi}_{b}\rangle$.
From such relations, we obtain

$\displaystyle\begin{split}&\langle\phi_{a}\phi_{b}\rangle=\frac{1}{2}R_{11}\delta_{ab},\\\
&\langle\phi_{a}\sigma_{b}\rangle=\frac{i}{2}R_{21}\delta_{ab},\\\
&\langle\sigma_{a}\phi_{b}\rangle=\frac{i}{2}R_{12}\delta_{ab},\\\
&\langle\sigma_{a}\sigma_{b}\rangle=-\frac{1}{2}R_{22}\delta_{ab}.\end{split}$
(98)

Here, because $\phi,\sigma$ are bosons, the second and the third relations
require $R_{12}=R_{21}$, which indeed holds in (95) and (96).

By putting (94) and (98) into (93), we obtain

$\displaystyle\begin{split}V_{\parallel\perp_{1},\perp_{2}}(R)=&-A_{1}\left(R_{22}{\bar{\psi}_{\parallel}}{\psi_{\parallel}}+R_{12}({\bar{\psi}_{\parallel}}{\varphi_{\parallel}}+{\bar{\varphi}_{\parallel}}{\psi_{\parallel}})+R_{11}{\bar{\varphi}_{\parallel}}{\varphi_{\parallel}}\right)\\\
&+A_{1}\left(R_{22}{\phi_{\parallel}}^{2}-2iR_{12}{\phi_{\parallel}}{\sigma_{\parallel}}-R_{11}{\sigma_{\parallel}}^{2}\right)\\\
&-A_{2}\left(R_{22}{\bar{\psi}_{\perp_{1}}}{\psi_{\perp_{1}}}+R_{12}({\bar{\psi}_{\perp_{1}}}{\varphi_{\perp_{1}}}+{\bar{\varphi}_{\perp_{1}}}{\psi_{\perp_{1}}})+R_{11}{\bar{\varphi}_{\perp_{1}}}{\varphi_{\perp_{1}}}\right)\\\
&+A_{2}\left(R_{22}{\phi_{\perp_{1}}}^{2}-2iR_{12}{\phi_{\perp_{1}}}{\sigma_{\perp_{1}}}-R_{11}{\sigma_{\perp_{1}}}^{2}\right),\end{split}$
(99)

where $A_{i}$ are given in (63). Adding (99) to (49), we obtain the full
kinetic terms of the $\parallel\perp_{1}$ fields. In particular the fermionic
part has the form,

$\displaystyle
K_{\parallel\perp_{1}}+V_{\parallel\perp_{1},\perp_{2}}(R)=\bar{\psi}_{\parallel\perp_{1}}H\psi_{\parallel\perp_{1}}+\hbox{bosonic
part},$ (100)

where
$\bar{\psi}_{\parallel\perp_{1}}=({\bar{\psi}_{\parallel}},{\bar{\varphi}_{\parallel}},{\bar{\psi}_{\perp_{1}}},{\bar{\varphi}_{\perp_{1}}}),\psi_{\parallel\perp_{1}}=({\psi_{\parallel}},{\varphi_{\parallel}},{\psi_{\perp_{1}}},{\varphi_{\perp_{1}}})$
and $H$ is given in (62). Because of the supersymmetry, the bosonic part has
essentially the same structure as $H$.

##  Appendix Appendix F Exact expression of $Z_{\perp_{2}}$

$Z_{\perp_{2}}$ can be taken from [26], because it is the same as the
partition function of the transverse components of the fields in [26], which
is denoted as $G_{N}$ there. A point to note is that, while the dimension of
the transverse directions there is $N-1$, the dimension of $\perp_{2}$ of this
paper is $N-2$. Therefore we have to deduct $N$ by one191919See also the
footnotes associated to (53) and (94)., when we take a result from [26]. In
our current case of $N=9$, this corresponds to $G_{N=8}$, and therefore

$\displaystyle\begin{split}Z_{\perp_{2}}^{N=9}&=G_{N=8}\\\
&=\pi^{\frac{13}{2}}\Bigg{(}\frac{\sqrt{2}e^{-\frac{1}{8z}}(1+210z^{2}-2100z^{3}+12600z^{4}+25200z^{5})}{15z^{\frac{3}{2}}}\\\
&\hskip
142.26378pt+(1-42z+420z^{2}-840z^{3})\,\gamma\left[\frac{1}{2},\frac{1}{8z}\right]\Bigg{)},\end{split}$
(101)

where $z=v^{2}/(6\alpha)$, and $\gamma\left[1/2,y\right]$ is the lower
incomplete gamma function with index $1/2$, which is related to the error
function by

$\displaystyle\gamma\left[\frac{1}{2},y\right]=\sqrt{\pi}\,\hbox{Erf}\left(\sqrt{y}\right).$
(102)

##  Appendix Appendix G Proposition 2 of [29]

In this appendix we compare our threshold value ${\tilde{\alpha}}$ with the
value given in Proposition 2 of [29].

In [29] with their notations, the random tensor $Y$ with a background is given
by

$\displaystyle\begin{split}Y=\lambda u^{\otimes
k}+\frac{1}{\sqrt{2N}}W,\end{split}$ (103)

where $|u|=1$, $W=\sum_{\pi}G^{\pi}/k!$ with $G_{i_{1}\cdots i_{k}}\sim
N(0,1)$, and $\pi$ denotes permutations. By taking $k=3$ and computing the
standard deviations of $W_{abc}$, we find

$\displaystyle W_{abc}\sim N\left(0,1/\sqrt{d_{abc}}\right),$ (104)

where $d_{abc}$ is the degeneracy factor defined in (67). In our case,
$C_{abc}\sim N(0,1/\sqrt{2\alpha d_{abc}})$, and therefore

$\displaystyle\alpha=N$ (105)

is taken in [29] in our notation. $q=\lambda$ is also taken. Therefore,

$\displaystyle\tilde{\alpha}=\frac{\alpha q^{2}}{N}=\lambda^{2}.$ (106)

Proposition 2 states

$\displaystyle\lambda_{c}^{2}=\frac{(k-1)^{k-1}}{2k(k-2)^{k-2}}=\frac{2}{3}$
(107)

for $k=3$. Therefore the threshold value of [29] corresponds to
${\tilde{\alpha}}_{c}=2/3$.

##  Appendix Appendix H Maximums of $S_{\infty}^{\gamma}$

In this section we will prove that $S_{\infty}^{\gamma}<0$ for
$0<{\tilde{v}}<\infty$ and $-1\leq{n_{\parallel}}<1$. Firstly,

$\displaystyle(v^{4}+4{\tilde{\alpha}}{\tilde{\beta}})(v^{4}+12{\tilde{\alpha}}{\tilde{\beta}})S_{\infty}^{\gamma}=-{\tilde{\alpha}}{\tilde{v}}^{2}({\tilde{v}}^{4}-2{n_{\parallel}}^{3}{\tilde{v}}^{5}+(3{n_{\parallel}}^{4}-2{n_{\parallel}}^{6}){\tilde{v}}^{6}+12{\tilde{\alpha}}{\tilde{\beta}}(1-2{n_{\parallel}}^{3}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2})).$
(108)

Therefore the statement is equivalent to prove the positivity of the quantity
in the parentheses:

$\displaystyle\begin{split}&{\tilde{v}}^{4}-2{n_{\parallel}}^{3}{\tilde{v}}^{5}+(3{n_{\parallel}}^{4}-2{n_{\parallel}}^{6}){\tilde{v}}^{6}+12{\tilde{\alpha}}{\tilde{\beta}}(1-2{n_{\parallel}}^{3}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2})\\\
&\
=\left(1-2{n_{\parallel}}^{3}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2}\right){\tilde{v}}^{4}+2{n_{\parallel}}^{4}(1-{n_{\parallel}}^{2}){\tilde{v}}^{6}+12{\tilde{\alpha}}{\tilde{\beta}}(1-2{n_{\parallel}}^{3}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2}).\end{split}$
(109)

For ${n_{\parallel}}<1$, we find

$\displaystyle
1-2{n_{\parallel}}^{3}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2}>1-2{n_{\parallel}}^{2}{\tilde{v}}+{n_{\parallel}}^{4}{\tilde{v}}^{2}=(1-{n_{\parallel}}^{2}{\tilde{v}})^{2}\geq
0.$ (110)

Similarly one can prove that the quantities in the other parentheses are
larger than zero.

## References

  * [1] E. P. Wigner, “On the Distribution of the Roots of Certain Symmetric Matrices,” Annals of Mathematics 67 (2), 325-327 (1958) https://doi.org/10.2307/1970008.
  * [2] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978) doi:10.1007/BF01614153
  * [3] B. Eynard, Counting surfaces : CRM Aisenstadt chair lectures, Prog. Math. Phys. 70, Birkhäuser, 2016.
  * [4] D. J. Gross and E. Witten, “Possible Third Order Phase Transition in the Large N Lattice Gauge Theory,” Phys. Rev. D 21, 446-453 (1980) doi:10.1103/PhysRevD.21.446
  * [5] S. R. Wadia, “$N$ = Infinity Phase Transition in a Class of Exactly Soluble Model Lattice Gauge Theories,” Phys. Lett. B 93, 403-410 (1980) doi:10.1016/0370-2693(80)90353-6
  * [6] J. Ambjorn, B. Durhuus and T. Jonsson, “Three-dimensional simplicial quantum gravity and generalized matrix models,” Mod. Phys. Lett. A 6, 1133-1146 (1991) doi:10.1142/S0217732391001184
  * [7] N. Sasakura, “Tensor model for gravity and orientability of manifold,” Mod. Phys. Lett. A 6, 2613-2624 (1991) doi:10.1142/S0217732391003055
  * [8] N. Godfrey and M. Gross, “Simplicial quantum gravity in more than two-dimensions,” Phys. Rev. D 43, R1749(R) (1991) doi:10.1103/PhysRevD.43.R1749
  * [9] R. Gurau, “Colored Group Field Theory,” Commun. Math. Phys. 304, 69-93 (2011) doi:10.1007/s00220-011-1226-9 [arXiv:0907.2582 [hep-th]].
  * [10] M. Ouerfelli, V. Rivasseau and M. Tamaazousti, “The Tensor Track VII: From Quantum Gravity to Artificial Intelligence,” [arXiv:2205.10326 [hep-th]].
  * [11] L. Qi, “Eigenvalues of a real supersymmetric tensor,” Journal of Symbolic Computation 40, 1302-1324 (2005).
  * [12] L.H. Lim, “Singular Values and Eigenvalues of Tensors: A Variational Approach,” in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP ’05), Vol. 1 (2005), pp. 129–132.
  * [13] D. Cartwright and B. Sturmfels, “The number of eigenvalues of a tensor,” Linear algebra and its applications 438, 942-952 (2013).
  * [14] L. Qi, H. Chen, Y, Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018.
  * [15] A. Crisanti and H.-J. Sommers, “The spherical p-spin interaction spin glass model: the statics”, Z. Phys. B 87, 341 (1992).
  * [16] T. Castellani and A. Cavagna, “Spin-glass theory for pedestrians”, J. Stat. Mech.: Theo. Exp. 2005, P05012 [arXiv: cond-mat/0505032].
  * [17] A. Auffinger, G.B. Arous, and J. Černý, “Random Matrices and Complexity of Spin Glasses”, Comm. Pure Appl. Math., 66, 165-201 (2013) doi.org/10.1002/cpa.21422 [arXiv:1003.1129 [math.PR]].
  * [18] A. Biggs, J. Maldacena and V. Narovlansky, “A supersymmetric SYK model with a curious low energy behavior,” [arXiv:2309.08818 [hep-th]].
  * [19] O. Evnin, “Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes,” Class. Quant. Grav. 38, no.20, 203001 (2021) doi:10.1088/1361-6382/ac1b46 [arXiv:2104.09797 [gr-qc]].
  * [20] P. Breiding, “The expected number of eigenvalues of a real gaussian tensor,” SIAM Journal on Applied Algebra and Geometry 1, 254-271 (2017).
  * [21] P. Breiding, “How many eigenvalues of a random symmetric tensor are real?,” Transactions of the American Mathematical Society 372, 7857-7887 (2019).
  * [22] O. Evnin, “Melonic dominance and the largest eigenvalue of a large random tensor,” Lett. Math. Phys. 111, 66 (2021) doi:10.1007/s11005-021-01407-z [arXiv:2003.11220 [math-ph]].
  * [23] R. Gurau, “On the generalization of the Wigner semicircle law to real symmetric tensors,” [arXiv:2004.02660 [math-ph]].
  * [24] N. Sasakura, “Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory,” Phys. Lett. B 836, 137618 (2023) doi:10.1016/j.physletb.2022.137618 [arXiv:2208.08837 [hep-th]].
  * [25] N. Sasakura, “Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory,” PTEP 2023, no.1, 013A02 (2023) doi:10.1093/ptep/ptac169 [arXiv:2209.07032 [hep-th]].
  * [26] N. Sasakura, “Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small $N$,” J. Math. Phys. 64, 063501 (2023) doi:10.1063/5.0133874 [arXiv:2210.15129 [hep-th]].
  * [27] A. Montanari, E. Richard, “A statistical model for tensor PCA,” Advances in Neural Information Processing Systems, 27 (2014).
  * [28] T. Lesieur, L. Miolane, M. Lelarge, F. Krzakala and L. Zdeborova, “Statistical and computational phase transitions in spiked tensor estimation”, in 2017 IEEE International Symposium on Information Theory (ISIT), pp. 511–515, IEEE, 2017.
  * [29] G. B. Arous, S. Mei, A. Montanari and M. Nica, “The landscape of the spiked tensor model”, Communications on Pure and Applied Mathematics 72 , no.11, 2282–2330 (2019).
  * [30] N. Delporte, N. Sasakura, R. Toriumi, in preparation.
  * [31] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, (Clarendon Press, Oxford, 1989).
  * [32] Y. V. Fyodorov, A. Nock, “On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials,” J. Stat. Phys. 159, 731-751 (2015). https://doi.org/10.1007/s10955-015-1209-x
  * [33] E. Witten, “An SYK-Like Model Without Disorder,” J. Phys. A 52, no.47, 474002 (2019) doi:10.1088/1751-8121/ab3752 [arXiv:1610.09758 [hep-th]].
  * [34] T. Kawano and N. Sasakura, “Emergence of Lie group symmetric classical spacetimes in the canonical tensor model,” PTEP 2022, no.4, 043A01 (2022) doi:10.1093/ptep/ptac045 [arXiv:2109.09896 [hep-th]].
  * [35] N. Sasakura, “Splitting-merging transitions in tensor-vectors systems in exact large-N limits,” Phys. Rev. D 106, no.12, 126016 (2022) doi:10.1103/PhysRevD.106.126016 [arXiv:2206.12017 [hep-th]].