We obtain exact analytic expressions of real tensor eigenvalue/vector distributions of real symmetric order-three tensors with Gaussian distributions for N ≤ 8. This is achieved by explicitly computing the partition function of a zero-dimensional boson–fermion system with four interactions. The distributions are expressed by combinations of polynomial, exponential, and error functions as results of feasible complicated bosonic integrals that appear after fermionic integrations. By extrapolating the expressions and also using a previous result, we guess a large-N expression. The expressions are compared with Monte Carlo simulations, and precise agreement and good agreement are obtained with the exact and the large-N expressions, respectively. Understanding the feasibility of the integration is left for future study, which would provide a general-N analytic formula.
REFERENCES
1.
E. P.
Wigner
, “On the distribution of the roots of certain symmetric matrices
,” Ann. Math.
67
(2
), 325
–327
(1958
).2.
E.
Brézin
, C.
Itzykson
, G.
Parisi
, and J. B.
Zuber
, “Planar diagrams
,” Commun. Math. Phys.
59
, 35
(1978
).3.
B.
Eynard
, Counting Surfaces: CRM Aisenstadt Chair Lectures
, Progress in Mathematical Physics Vol. 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
).5.
S. R.
Wadia
, “N = ∞ phase transition in a class of exactly soluble model lattice gauge theories
,” Phys. Lett. B
93
, 403
–410
(1980
).6.
J.
Ambjørn
, B.
Durhuus
, and T.
Jónsson
, “Three-dimensional simplicial quantum gravity and generalized matrix models
,” Mod. Phys. Lett. A
6
, 1133
–1146
(1991
).7.
N.
Sasakura
, “Tensor model for gravity and orientability of manifold
,” Mod. Phys. Lett. A
6
, 2613
–2624
(1991
).8.
N.
Godfrey
and M.
Gross
, “Simplicial quantum gravity in more than two-dimensions
,” Phys. Rev. D
43
, R1749(R)
(1991
).9.
R.
Gurau
, “Colored group field theory
,” Commun. Math. Phys.
304
, 69
–93
(2011
); 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
, H.
Chen
, and Y.
Chen
, Tensor Eigenvalues and Their Applications
(Springer
, Singapore
, 2018
).12.
P.
Breiding
, “The expected number of eigenvalues of a real Gaussian tensor
,” SIAM J. Appl. Algebra Geom.
1
, 254
–271
(2017
).13.
P.
Breiding
, “How many eigenvalues of a random symmetric tensor are real?
,” Trans. Am. Math. Soc.
372
, 7857
–7887
(2019
).14.
O.
Evnin
, “Melonic dominance and the largest eigenvalue of a large random tensor
,” Lett. Math. Phys.
111
, 66
(2021
); arXiv:2003.11220 [math-ph].15.
R.
Gurau
, “On the generalization of the Wigner semicircle law to real symmetric tensors
,” arXiv:2004.02660 [math-ph].16.
N.
Sasakura
, “Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory
,” Phys. Lett. B
836
, 137618
(2023
); arXiv:2208.08837 [hep-th].17.
N.
Sasakura
, “Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory
,” Prog. Theor. Exp. Phys.
2023
(1
), 013A02
; arXiv:2209.07032 [hep-th].18.
A.
Crisanti
and H.-J.
Sommers
, “The spherical p-spin interaction spin glass model: The statics
,” Z. Phys. B: Condens. Matter
87
, 341
(1992
).19.
T.
Castellani
and A.
Cavagna
, “Spin-glass theory for pedestrians
,” J. Stat. Mech.: Theo. Exp.
2005
, P05012
; arXiv:cond-mat/0505032.20.
A.
Auffinger
, G. B.
Arous
, and J.
Černý
, “Random matrices and complexity of spin glasses
,” Commun. Pure Appl. Math.
66
, 165
–201
(2013
); arXiv:1003.1129 [math.PR].21.
L.
Qi
, “Eigenvalues of a real supersymmetric tensor
,” J. Symbolic Comput.
40
, 1302
–1324
(2005
).22.
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)
(IEEE
, 2005
), Vol. 1
, pp. 129
–132
.23.
D.
Cartwright
and B.
Sturmfels
, “The number of eigenvalues of a tensor
,” Linear Algebra Appl.
438
, 942
–952
(2013
).24.
Y. V.
Fyodorov
and A.
Nock
, “On random matrix averages involving half-integer powers of GOE characteristic polynomials
,” J. Stat. Phys.
159
, 731
–751
(2015
).25.
J.
Zinn-Justin
, Quantum Field Theory and Critical Phenomena
(Clarendon Press
, Oxford
, 1989
).26.
27.
T.
Kawano
and N.
Sasakura
, “Emergence of Lie group symmetric classical spacetimes in the canonical tensor model
,” Prog. Theor. Exp. Phys.
2022
(4
), 043A01
; arXiv:2109.09896 [hep-th].28.
N.
Sasakura
, “Splitting-merging transitions in tensor-vectors systems in exact large-N limits
,” Phys. Rev. D
106
(12
), 126016
(2022
); arXiv:2206.12017 [hep-th].29.
A.
Crisanti
, L.
Leuzzi
, and T.
Rizzo
, “The complexity of the spherical p-spin spin glass model, revisited
,” Eur. Phys. J. B
36
, 129
–136
(2003
); arXiv:cond-mat/0307586.© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
