Real eigenvector distributions of random tensors with backgrounds and random deviations

As in random matrix theories, eigenvector/value distributions are important 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.


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 eigenvalue1 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 limit 2 .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 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 [27][28][29].We also introduce random deviations to eigenvector equations 3 .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.

Real tensor eigenvector equation with backgrounds and deviations
In this paper we restrict ourselves to order-three tensors 4 for simplicity.We consider the following eigenvector equation [11][12][13][14] with a background tensor Q and a deviation vector η, Here the indices take a, b, c = 1, 2, . . ., 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, η are real vectors: While Q is an externally given background tensor, C abc is a random tensor with Gaussian distribution of a zero mean value.The vector η 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 η, the setup goes back to the cases previously studied in [24][25][26].
For given Q, C, η, the distribution of v is given by where v i (i = 1, 2, . . ., #sol(Q, C, η)) are all the real solutions to (1), and the absolute value of the determinant of the matrix, which is the Jacobian factor associated to the change of the variables of the delta functions in (3).
When C, η have Gaussian distributions with zero mean values, the eigenvector distributions are computed by taking the average over C, η: where α, β > 0, #C is the number5 of the independent components of . Here a slightly complicated introduction of β is for later convenience.By using the well known formula, 1 2π R dλ e iλx = δ(x), the integration of the delta functions over η in (5) can be rewritten as Therefore, by putting this into (5), we obtain The part |det M (v, Q, C)| 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.

Quantum field theory expression
The quantity we will compute in this section is defined by ignoring taking the absolute value in (7): Following backward the derivation in Section 2, the distribution corresponds to a "signed" distribution, which has an extra sign of detM (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, ψa , ψ a (a = 1, 2, This technique to incorporate determinants in quantum field theories is common in treating disordered systems in statistical physics 6 .Then (8) can be rewritten as where Since C and λ appear at most quadratically in (11), they can be integrated out by Gaussian integrations.We will first integrate over C and then over λ.Though the integrations are straightforward, the actual computation is a little cumbersome, because of the anticommuting 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 η.Now, new terms in S signed bare compared to [24] are those depending on Q and β, and are explicitly given by 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 λ among the generated ones, iλ a v a in (11), and the terms depending on λ in (12), we obtain the λ-dependent part of the action as where , and D signed can be taken from [24] 7 , 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 where I ∥ and I ⊥ are the projection matrices to the parallel and the transverse subspaces against v: Then the integration over λ with the action (13) generates an action, where the inverse of B is given by When we consider the case with Q = β = 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 ̸ = 0 or β ̸ = 0.By subtracting δS signed λ for Q = β = 0 in (16) and using (17), we obtain where [24] is given by where with (19) and the last term in ( 12) to ( 21) and doing some straightforward computations, we finally obtain where Some details of the derivation are explained in Appendix A.

Rank-one Q
To study the formula ( 22) with ( 23) more explicitly, let us consider the case that Q is a rank-one tensor, 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 (namely, v • n 1 = 0, |n 1 | = 1).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 ⊥ 2 .Then the transverse fermions, ψ⊥ , ψ ⊥ , can further be decomposed into ψ⊥ 1 = n 1 • ψ and ψ⊥ 2 = I ⊥ 2 ψ and similarly for ψ ⊥ .Note that ψ⊥ For (24), We also notice Putting these into ( 22) and ( 23), we obtain where It is not difficult to explicitly compute the fermion integration in (26).As is shown in Appendix B, we obtain where U denotes the confluent hypergeometric function of the second kind, and b i , d i are the coefficients of the terms in ( 27): The result ( 26) with (28) gives the exact expression of the signed distribution.

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 where I is an identity matrix of N -by-N , ϕ a , σ a are real bosons, ψa , ψ a , φa , φ a are fermions, and ψψ = ψa ψ a , etc.Here we have introduced a positive infinitesimal parameter ϵ 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 where As in Section 3, there are no new terms depending on C compared with the previous case for [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 λ as in Section 3: where B, D Q are already defined in (15) and below (13), respectively.Here D can be taken from [26] 8 : where va = v a /|v|.Comparing ( 33) with ( 13), the change is to replace D signed with −D.By using (19) with this replacement and adding the Q-dependent but λ-independent terms in (32), we obtain where Z is a partition function of a quantum field theory, Here S 0 is the former result in [26] corresponding to Q = β = 0, which is explicitly given in Appendix C, and where Note that the first three terms are some corrections to the kinetic terms, and the latter to the four-interaction terms.As for D ∥ and D ⊥ , we have more explicit expressions from (34), The four-interaction terms in (37) have the form of self-products.One can make it quadratic by using the formula The result is where g ∥ is one dimensional, g ⊥ is N − 1 dimensional, and 9 which contains only quadratic terms of the fields.

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).

A general formula
By putting ( 24) into (35), one obtains 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 As for (40), we obtain In the following subsections, we will consider N = 1, N = 2 and large-N cases.

N = 1
In this case we ignore all the transverse components, and also set n ∥ = 1.By putting these to (35), (39), ( 43) and (C2), and doing some straightforward computations, we obtain where The details of the derivation are given in Appendix D.

N = 2
In this case the transverse direction is exhausted by one-dimension, namely, ⊥=⊥ 1 and ⊥ 2 is null.A special fact about this case is that the four-interaction terms in (C2) have a form of a square: Therefore we can rewrite this part of the action as whose exponent contains only quadratic terms of the fields.Using this for (39), ( 43) and (C2), we obtain where with Then the integration (48) over the fields generates a square root of a determinant, and we obtain

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 ⊥ 2 fields will dominate over those of the ∥⊥ 1 fields, since the former is (N − 2)-dimensional, while the latter is 2-dimensional.Therefore the dynamics of the ⊥ 2 fields can well be determined by themselves with little effects from the ∥⊥ 1 fields, which may be ignored in the large-N limit.Then the dynamics of the ∥⊥ 1 fields may be computed in the backgrounds of the ⊥ 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 Here Z ⊥ 2 is the partition function determined solely by the ⊥ 2 fields, where S ⊥ 2 is the collection of the terms which contain only the ⊥ 2 fields in (C1) with (C2) 10 .
The computation of the partition function Z ⊥ 2 is the same as that in the previous paper [26], because S ⊥ 2 has the same form as the action of the transverse directions there 11 .
Z ∥⊥ 1 (R) is the partition function of the ∥⊥ 1 fields in the background of the ⊥ 2 fields, where R denotes the classical backgrounds of the ⊥ 2 fields, as will be explained below in more detail.Here the action S ∥⊥ 1 (R) is composed of all the terms which contain the ∥⊥ 1 fields in (42) and (C1).Part of the terms in S ∥⊥ 1 (R) contain the ⊥ 2 fields as well.For large N these ⊥ 2 fields may well be approximated by their classical values because of the large degrees of freedom of the ⊥ 2 fields.For instance, we perform replacements, where ⟨•⟩ denotes an expectation value.By doing such replacements we obtain S ∥⊥ 1 (R), whose dynamical fields are only the ∥⊥ 1 fields. 10For instance, we include ψ⊥2 , because of the reason mentioned in the first paragraph.The ignored terms will be considered in Z ∥⊥1 .
11 But note the difference of the dimensions of ⊥ 2 here and ⊥ 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.
Obtaining the explicit form of S ∥⊥ 1 (R) proceeds as follows.The quadratic and quartic terms of the ∥⊥ 1 fields can be processed in the same manner as are performed for N = 2 in Section 4.2.3, and we obtain K ∥⊥ 1 in (49) with (50).Then the four-interaction terms between the ∥⊥ 1 fields and the ⊥ 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 (E7) of Appendix E. Thus we have whose terms are all quadratic in the ∥⊥ 1 fields.Then the computation of the partition function ( 54) is just a computation of a determinant, and we obtain where H is given by where a i are given in (50), R ij are the values of the two point of correlation functions of the ⊥ 2 fields explicitly given in (E3) and (E4), and The derivation of H is given in Appendix E.

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].Throughout this section we put α = 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 η.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 ≲ 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.(60) • As explained above, compute all the complex solutions to the eigenvector equation ( 1), and pick up only the real ones • Repeat the above processes.
By this sampling procedure, we obtain a series of data, , where L denotes the total number of real solutions obtained 13 .
To plot the distributions, we classify the data into equally spaced bins in v and angle θ where v, θ are the center values of a bin, and δ v , δ θ are the sizes of a bin.We denote the total number of data satisfying (61) as N δv,δ θ ,+ (v, θ) and N δv,δ θ ,− (v, θ) for Then the distribution of the real eigenvectors from a data is given by where N M C denotes the total number of sampling processes in obtaining the data and the ± part represents error estimates.The signed distribution is given by As for the analytical side, since we take only the size |v| and the relative angle θ as data, the above MC distributions should be compared with where S N −2 = 2π (N −1)/2 /Γ[(N − 1)/2] is the surface volume of a unit sphere in the N − 1dimensional flat space.Here ρ(v, q, n, β) is one of the expressions obtained in Sections 3 and 4, and v in the argument of ρ on the righthand side abusively denotes an arbitrary vector In the following we will compare the Monte Carlo and the analytical results.Let us first consider the signed distribution.The analytical result is obtained by putting ( 26) with ( 28) into (64).Since the analytical result is an exact result, it should agree with the MC result within errors.In Figure 1 As in Figure 1 and the left slot of Figure 2, an evident negative peak can be observed around |v| ∼ 0.1 and θ ∼ 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| ∼ q −1 , while the angle is not strictly θ = 0.The reason is that the volume factor in (64) contains sin N −2 (θ), and pushes the peak away from θ = 0.Because of the same reason, the other major structures are concentrated around θ = π/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| ∼ 0.04, θ ∼ π/2.This peak corresponds to the trivial eigenvector v = 0.Because of β > 0 the distribution broadens around |v| ∼ 0, and the volume factor v N −1 in (64) pushes the peak away from |v| = 0.
In Figure 3 the MC distribution (62) 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 Fig. 3 The MC distribution ( 62) is plotted for the same data as used in Figure 1  expression of Z ⊥ 2 can be taken from the previous result in [26], which is explicitly given in Appendix F. As for Z ∥⊥ 1 , by numerically integrating (57) on a grid of points in |v| and θ, an interpolation function of Z ∥⊥ 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.

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, αq 2 /N ≳ 0.6, βq 2 N ≲ 0.1.In particular for βq 2 = 0, we will find the threshold value to be 0.66 < (αq 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 ∥ = 0 and therefore relatively vanishes in the strict large-N limit.In the other scaling limit, αq 2 ∼ N γ , βq 2 ∼ N −γ with γ > 1, the peak remains in the strict large-N limit.
We want to consider large-N limits which keep both the parameters Q and β relevant.
As was discussed in Section 5, the volume factor sin N −2 θ in (64) 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 θ ∼ e N log(sin θ) .A large-N scaling which makes the exponential factor in (41) in this order is given by where α, β 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 ⊥ 2 Z ∥⊥ 1 in Section 4.2.4.The large-N limit of Z ⊥ 2 was determined in [25], and it is given by where 14 with 15 x = (N − 2)v 2 /(3α) ∼ ṽ2 /(3α).As for Z ∥⊥ 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 (59) and R ij in (E3) and (E4) drop out.Therefore H does not depend on g i and we get which has no relevant effects to the formula below for the large N limit. 14For simplicity, S ∞ ⊥2 is shifted by an irrelevant constant from the corresponding expression with R = 1/2 in [25]. 15N must be deducted by one, when we take a result from [25].See a footnote below (E2). -3 Fig. 5 In the shaded region of the parameters, the eigenvector distribution has a peak of S ∞ corresponding to the eigenvector of Q.
By collecting the results above and using ( 64) and (41), we obtain where n ∥ = cos θ (n ⊥ = sin θ), and const. is the part not dependent on ṽ or θ.
It is interesting to study the profile of S ∞ (ṽ, θ) in the ṽ and θ plane for various values of α, β.We have numerically studied it for the parameter region 10 −3 ≤ α ≤ 10 3 , 10 −3 ≤ β ≤ 10 3 .In the unshaded region of Figure 5, the peak(s) exist only along n ∥ = 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 ∥ = 0, there exists also a peak which has non-zero n ∥ .This peak corresponds to the eigenvector q −1 n of the background tensor Q, as is shown in the right slot of Figure 6 as an example.In Figure 7, the values of n ∥ and ṽ are plotted for the latter peak.The location can be well identified with q −1 n, if the values take n ∥ ∼ 1 and ṽ ∼ 1.As can be seen in the plots, this occurs in the region, log 10 α ≳ −0.2 and log 10 β ≲ −1.This is the parameter region in which the background tensor Q can be detected well.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 α ∼ −0.2, namely, α ∼ 0.63, independent of β for log 10 β ≲ −1.To see the threshold value more precisely for β = 0, we plot n ∥ and ṽ of the peak with n ∥ > 0 in Figure 8.We find that the peak does not exist at α ≤ 0.66, but exists at α ≥ 0.67 with n ∥ ≳ 0.7.On the other hand, as explained in Appendix G,  Proposition 2 of [29] states that the threshold value is α = 2/3, which indeed agrees with our value.
We numerically observed that a peak at n ∥ = 0 always take the largest value of S ∞ at least in the parameter region of α, β we have studied above.This means that, because ρ ∼ e N S∞ , the peak corresponding to Q will effectively be invisible compared to the peak(s) at n ∥ = 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).
The main reason for the above difficulty of detection comes from the strong effect of the volume factor sin N −2 θ in (64), which enhances the region n ∥ ∼ 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 In this limit, x = (N − 2)v 2 /(3α) ∼ N −γ+1 → 0 in the large-N limit, so therefore (67) becomes a constant, meaning that Z ⊥ 2 is a free theory independent of v.As for Z ∥⊥ 1 , A i → 0 and R ij approaches finite values, so Z ∥⊥ 1 is again a finite quantity.Therefore from (41), the major contribution comes only from the exponent, and we obtain As is shown in Appendix H, it is straightforward to prove that the maximum value of S γ ∞ is 0, and this occurs only at three locations: (i) ṽ = 0, (ii) ṽ → ∞, n ∥ = 0, (iii) ṽ = 1, n ∥ = 1.The last location corresponds to the background Q.An example of S γ ∞ is shown in Figure 9. Since the eigenvector distribution is given by ρ ∼ e N γ S γ ∞ , there remains only the three locations above in the strict large-N limit.This means that, in the limit, a finite eigenvector (v ̸ = 0, ∞) is surely that of the background Q.

Summary and future prospects
In this paper we have studied the real eigenvector distributions of real symmetric orderthree 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, exactly or approximately, 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 [6][7][8][9], 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 tensor models 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.
A Derivation of ( 22) From ( 14), the parallel/transverse parts of D signed are given by By putting (A1) into (19), we obtain Adding this and the last term of ( 12) to ( 21), one obtains (22) with (23).
B Derivation of (28) Let us parametrize (27) as follows: Then by explicitly performing the fermion integrations for ∥ and ⊥ 1 directions, we obtain Now the last fermion integration can be computed as 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: where the hypergeometric function has a formal series expansion, with the Pochhammer symbol, (a) n = a(a + 1) • • • (a + n − 1) ((a) 0 = 1).For the argument in (B3), the formal series stops at finite n, and hence (B4) is an exact relation.One can also find the confluent hypergeometric function here can be expressed by an hermite polynomial.
C Explicit form of S 0 The Q = β = 0 case was studied in [26], and the action for this case is given by where 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.
The boson-fermion integration in (D1) produces a square root of the determinant of a two-by-two matrix.It is easy to see that the ϵ → +0 limit is smooth, and we obtain

E Interactions between the ∥⊥ 1 and ⊥ 2 fields
There are no quadratic terms containing one ∥⊥ 1 field and one ⊥ 2 field, because the index of the ⊥ 2 field cannot be contracted with v or n.Therefore the ∥⊥ 1 fields can couple with the ⊥ 2 fields only through the four-interaction terms in (42) and (C2).By noting that for arbitrary fields X, Y , and collecting all the interaction terms between the ∥⊥ 1 and the ⊥ 2 fields, we obtain The expectation values of the ⊥ 2 fields can be taken from the large-N Schwinger-Dyson analysis performed in [25].The results were 16   ⟨ ψ⊥ where, with a newly introduced parameter 17 x = v 2 (N − 2)/(3α), 16 To 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 explained below, which assures the two-boson expectation values are just the copies of those of fermions. 17As the dimension of ⊥ 2 is N − 2, the formula presented in [25] for the dimension N − 1 of ⊥ must be replaced with N − 2.
• 0 < x < 1/4 2 ), 2 ). (E4) 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,β from (C1) and (37) are invariant under the following supersymmetry transformation: By putting (E2) and (E6) into (E1), we obtain

•
Randomly sample C and η.Each η a is randomly sampled by the normal distribution with the mean value zero and the standard deviation √ 2β.Each C abc is randomly sampled by the normal distribution with the mean value zero and the standard deviation 1/ √ d abc , corresponding to α = 1/2, where d abc is the degeneracy factor defined by 12 d abc = a = b = c, 3 for a ̸ = b = c or b ̸ = c = a or c ̸ = a = b, 6 for a ̸ = b ̸ = c ̸ = a.

Fig. 2
Fig. 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 θ; the left is at |v| = 0.105 and the right is at θ = π/2.
, we plot the MC result (63) for N = 9, β = 10 −4 , q = 10 with N M C = 4 • 10 4 .As examples, the analytical and MC results are compared at two slices, one at |v| = 0.105 and the other at θ = π/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.

Fig. 6
Fig. 6 In the left slot, S ∞ (const.being ignored) is plotted for log 10 α = −1, log 10 β = 1, which is in the unshaded region of Figure 5.The right slot is for log 10 α = 1, log 10 β = −2 in the shaded region.In the latter case, a peak near ṽ ∼ 1, θ ∼ 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 (67), where the function is continuous but its first derivative is discrete.

Fig. 7
Fig. 7 The values of n ∥ (left) and ṽ(right) of the peak with n ∥ > 0, corresponding to the eigenvector of Q, are plotted.Identification of this peak with Q can well be done in the region log 10 α ≳ −0.2 and log 10 β ≲ −1.

Fig. 9
Fig. 9 S γ∞ is plotted for log 10 α = 0 and log 10 β = −1 as an example.There is a peak corresponding to the eigenvector of the background tensor Q.