Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-Fermi theory. (English) Zbl 1518.81083
Summary: 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.
MSC:
| 81T32 | Matrix models and tensor models for quantum field theory |
| 60G15 | Gaussian processes |
| 60E05 | Probability distributions: general theory |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 65C05 | Monte Carlo methods |
References:
| [1] | Wigner, E., On the distribution of the roots of certain symmetric matrices, Ann. Math., 67, 2, 325-327 (1958) · Zbl 0085.13203 |
| [2] | Brezin, E.; Itzykson, C.; Parisi, G.; Zuber, J. B., Planar diagrams, Commun. Math. Phys., 59, 35 (1978) · Zbl 0997.81548 |
| [3] | Gross, D. J.; Witten, E., Possible third order phase transition in the large N lattice gauge theory, Phys. Rev. D, 21, 446-453 (1980) |
| [4] | Wadia, S. R., N = infinity phase transition in a class of exactly soluble model lattice gauge theories, Phys. Lett. B, 93, 403-410 (1980) |
| [5] | Ambjorn, J.; Durhuus, B.; Jonsson, T., Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A, 6, 1133-1146 (1991) · Zbl 1020.83537 |
| [6] | Sasakura, N., Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A, 6, 2613-2624 (1991) · Zbl 1020.83542 |
| [7] | Godfrey, N.; Gross, M., Simplicial quantum gravity in more than two-dimensions, Phys. Rev. D, 43, 1749-1753 (1991) |
| [8] | Gurau, R., Colored group field theory, Commun. Math. Phys., 304, 69-93 (2011) · Zbl 1214.81170 |
| [9] | Breiding, P., The expected number of eigenvalues of a real Gaussian tensor, SIAM J. Appl. Algebra Geom., 1, 254-271 (2017) · Zbl 1365.65092 |
| [10] | Breiding, P., How many eigenvalues of a random symmetric tensor are real?, Trans. Am. Math. Soc., 372, 7857-7887 (2019) · Zbl 1431.15023 |
| [11] | Evnin, O., Melonic dominance and the largest eigenvalue of a large random tensor, Lett. Math. Phys., 111, 66 (2021) · Zbl 1509.15004 |
| [12] | Gurau, R., On the generalization of the Wigner semicircle law to real symmetric tensors |
| [13] | Qi, L., Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40, 1302-1324 (2005) · Zbl 1125.15014 |
| [14] | Lim, L. H., Singular values and eigenvalues of tensors: a variational approach, (Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP ’05), vol. 1 (2005)), 129-132 |
| [15] | Cartwright, D.; Sturmfels, B., The number of eigenvalues of a tensor, Linear Algebra Appl., 438, 942-952 (2013) · Zbl 1277.15007 |
| [16] | Crisanti, A.; Sommers, H.-J., The spherical p-spin interaction spin glass model: the statics, Z. Phys. B, 87, 341 (1992) |
| [17] | Castellani, T.; Cavagna, A., Spin-glass theory for pedestrians, J. Stat. Mech. Theory Exp., Article 05012 pp. (2005) · Zbl 1456.82490 |
| [18] | Auffinger, A.; Arous, G. B.; Černý, J., Random matrices and complexity of spin glasses, Commun. Pure Appl. Math., 66, 165-201 (2013) · Zbl 1269.82066 |
| [19] | Crisanti, A.; Leuzzi, L.; Rizzo, T., The complexity of the spherical p-spin spin glass model, revisited, Eur. Phys. J. B, 36, 129-136 (2003) |
| [20] | Zinn-Justin, J., Quantum Field Theory and Critical Phenomena (1989), Clarendon Press: Clarendon Press Oxford |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
