Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by $\pm 1$, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-$N$ limits are taken with the characteristic oscillatory behavior of the formula being preserved.


Introduction
Eigenvalue distributions are one of the main tools in computations of matrix models [1,2], and are also useful for qualitative understanding of the dynamics through their topological properties [3,4]. It would be an interesting problem to study similar distributions in tensor models [5,6,7,8].
Though it is not difficult to numerically compute eigenvalues/vectors for a given tensor of a small size by using commonly-used computers, analytical understanding of their properties and large-N limits (thermodynamic limits) for ensembles of tensors are still very limited: In [9,10] the expected numbers of real tensor eigenvalues are computed; In [11] the largest eigenvalue of a typical tensor in a Gaussian ensemble is estimated; in [12] the Wigner semicircle law in matrix models is extended to tensor models. In this paper, we give an explicit formula for real tensor eigenvector distributions with signs for the simplest case. The formula is derived by computing a partition function of a four-fermi theory.
As the simplest case, we restrict ourselves to a real symmetric tensor of order-three, C abc (C abc = C bac = C bca ∈ R, a, b, c = 1, 2, . . . , N ). There exist various definitions of tensor eigenvalues/vectors [13,14,15]. In this paper, we employ a definition of real eigenvectors, Note that repeated indices are assumed to be summed over throughout this paper. Real eigenvalues h accompanied with real eigenvectors (Z-eigenvalues in [13]) can be deduced by normalizing v as w = v/|v|, with h = 1/|v|.
The distribution of v a for each case of C abc is given by where the matrix M (v) has components, and det denotes the matrix determinant. The determinant factor is to make each solution identically contribute under the measure dv = N a=1 dv a . In fact, where v i (i = 1, 2, . . . , #real sol.(C)) denote all the real solutions to (1) for a given C.
The eigenvector equation (1) can be considered to be a stationary point equation of a potential V = v a v a /2 − C abc v a v b v c /3. Then the matrix M (v) is a Hessian matrix at the stationary point.
An interesting quantity is the mean distribution of v under a Gaussian distribution of C: where C 2 = C abc C abc , α is a positive constant, and A = dC exp(−α C 2 ) is a normalization factor. Since the integration over It would not be straightforward to compute (7). We rather consider a more tractable quantity,ρ in the rest of this paper. The difference from (7) is that taking the absolute value has been ignored. Therefore,ρ(v) is the quantity, with an additional sign factor compared to (6). We may call it a signed eigenvector distribution because of this sign factor.
There do not seem to exist any apparent quantitative relations between ρ andρ, butρ can be used as a lower bound of ρ, since We can also expect a similar relation, to hold between a signed Z-eigenvalue distributionη, and a Z-eigenvalue distribution η. Here, using the relation (3) and thatρ(v) depends only on |v|, the signed Z-eigenvalue distribution is defined byη where S N −1 = 2π N/2 /Γ(N/2) is the surface area of the unit hypersphere in N -dimensions.
A potential mathematical interest of the signed distribution above could be found in relation with Morse theory. The potential V above may be regarded as a random Morse function, and summations of signs over stationary points are related to Euler characteristics. Thereforẽ ρ(v) will reflect some topological aspect over sections of constant |v| in terms of random Morse functions.
A more direct physical application of the signed distribution can be found in the context of spin glasses. The Hamiltonian of the spherical p-spin model [16,17] with p = 3 is defined by where w is the dynamical variable, and C is a random external field. The Hamiltonian has in general a macroscopic number of stationary points, which is called complexity, and it is important to know distributions of local minimums to understand the dynamics. By implementing the constraint in (13) by the method of Lagrange multiplier, one can find that the stationary points are given by the solutions to (2) with ±h being the energy, and the sign of M corresponds to (−1) #negative+1 , where #negative is the number of unstable directions around a stationary point when the energy is negative 1 . Thus the signed distribution can provide a good estimate of the energy range where local minimums dominate by looking at the negative value region ofρ(v) near the smallest end of |v|. We would also like to add that the present paper is closely related to the study [18], which counts the stationary points of the Hamiltonian of the spherical p-spin model in the large-N (thermodynamic) limit by using random matrix theory 2 .
2 A four-fermi theory (8) can be recast into a more tractable form by introducing some virtual variables. As well known, the determinant can be rewritten by fermionic variables by using dψdψ eψ aKabψb = det K [20], and the delta function by a bosonic variable: where the action S is given by and λ andψ, ψ are respectively bosonic and fermionic. Below we will integrate over the bosonic variables to finally obtain a fermionic theory, assuming that this change of the order of the integrations does not affect the final result.
Let us first integrate over C. The part containing C in (15) is given by Performing the Gaussian integration over C results in the cancellation of the prefactor A −1 in (14), and a change of the action by where σ denotes summation over all the permutations of the indices, a, b, c. Expanding the expression in (17), we obtain where A · B = A a B a . To derive this expression, we have usedψ ·ψ = ψ · ψ = 0, which follow from the fermionic property.
Let us next perform the integration over λ. Picking up the terms containing λ in (15) (with no C) and (18), we have where The inverse of B is straightforwardly determined to be Therefore the change of the action by the Gaussian integration over λ is obtained as and a multiplicative factor (12πα) As for the second term, we obtain after a straightforward computation shown in Appendix A. To compute the last term in (22), Therefore, where we have used (ψ · v) 2 = 0 (or (ψ · v) 2 = 0) because of the fermionic property. Putting (23), (24), and (26) into (22), we obtain Adding the remaining terms in (15) and (18) to (27), and taking into account the generated multiplicative factors, we finally obtain an expression ofρ via a four-fermi theory, with

An explicit expression ofρ(v)
To further compute (28), it is more convenient to separateψ, ψ into the parallel and transverse directions against v:ψ Note that ψ ⊥ , ψ ⊥ have N − 1 independent components, whileψ , ψ have one. With this decomposition, the action (29) can be rewritten as where we have usedψ 2 = ψ 2 = 0. Here we note that the transverse and parallel components are decoupled.
The parallel component has no interactions, and the integration generates a prefactor −1.
To compute the integral over the transverse components, let us first recall where (a) n = a(a + 1) · · · (a + n − 1) denotes the Pochhammer symbol. Note that (32) are non-zero only if 2n ≤ N − 1. Now let us compute the part withψ ⊥ , ψ ⊥ in (28) with (31) by expanding its interaction term: where · denotes the floor function, we have used a formula, and U is a confluent hypergeometric function of the second kind, which has a relation, in terms of the confluent hypergeometric function of the first kind, 1 F 1 . This finally leads to a compact expression,

Comparisons with numerical simulations
For a given general value of C, one can numerically compute the eigenvectors defined in (1) by an appropriate numerical method which solves systems of polynomial equations. 3 Generally such a method gives complex solutions as well, and we only pick up real eigenvectors to adjust to our interest. We used Mathematica 12 to solve the equations. We took the following processes for our numerical simulations.
• Generate C by the normal distribution: Each independent component is generated by Here σ is a random number following the normal distribution of mean value zero and standard deviation one. d(i, j, k) is a degeneracy factor defined by This corresponds to the distribution of C in (7) with α = 1/2, since due to C being a symmetric tensor.
• Compute all the real eigenvectors.
• Store the pair of the size |v| and the sign s of det M (v) for each eigenvector.
• Repeat the above processes.
By the above procedure we obtain a data set of (|v i |, s i ) (i = 1, 2, . . . , L), where L is the total number of data. Then we definẽ where N C denotes the total number of randomly generated C, δv is a bin size, k = 0, 1, 2, . . ., and θ is a support function which takes 1 if the inequality of the argument is satisfied, but zero otherwise. Then (39) is the numerical quantity corresponding toρ is the surface area of the unit hypersphere in N -dimensions. As shown in Figure 1, we obtain good agreement between the analytical and numerical results.
By ignoring the signs s i of the same data, we can also consider which corresponds to the distribution of real eigenvectors (with no signs). As for the inequality (10), the bound becomes looser for larger N , while it remains tight near the lowest end of the distribution, as shown in Figure 2.

Large-N limits
The formulaρ(v) in (36) has oscillatory behavior and the periods become smaller as N becomes larger. Since this is a characteristic structure, we will keep the oscillatory behavior in the large-N limit. For this purpose, it turns out that we should perform the following scaling, with fixedα.
With the above scaling, we obtain, for large-N , where J denotes the Bessel function of the first kind. To derive this asymptotic expression, we have used the Stirling's approximation and the following properties of the hypergeometric functions, From (42), we obtain for N → ∞, (44)

Summary and future prospects
In this paper, we have derived an explicit formula for signed distributions of real tensor eigenvectors for random real symmetric order-three tensors with Gaussian distributions as the simplest case. The formula is expressed in terms of the confluent hypergeometric function of the second kind, which has been derived by computing a partition function of a four-fermi theory. We have also discussed the tightness/looseness of the formula as lower bounds of real eigenvector distributions, and the large-N limit with the characteristic oscillatory behavior of the formula being preserved.
It would be interesting to extend the results in some directions. One is to consider other types of tensors. The tensors employed in colored tensor models [8] would be an especially interesting case, because of the presence of 1/N expansions in the models. Another direction would be to consider more complicated distributions than Gaussian for tensors, and study the responses to the signed distributions. These extended studies will reveal the interplays between dynamics and the signed distributions, definining their roles in tensor models. It will also be interesting to apply the present result to the spherical p-spin model for spin glasses, using the connection explained in Section 1.