Random Matrices and Holographic Tensor Models

TL;DR

This work extends the random-matrix perspective of holographic tensor models by analyzing the simplest uncolored tensor model (n=3, d=3) and showing that its spectral properties—density of states, level spacing, and spectral form factor—closely mirror those found in colored models and SYK. Through explicit construction of discrete symmetries (spectral mirror, time-reversal, fermion-parity, and others), the authors account for a 16-fold degeneracy and demonstrate a Bott-periodic structure in the symmetry classes as a function of n. They systematically categorize the random-matrix ensembles controlling both uncolored and Gurau-Witten models within the Andreev-Altland-Zirnbauer framework, revealing richer symmetry structures than the conventional Wigner-Dyson classes. The findings reinforce the view that holographic tensor models exhibit chaotic, black-hole-like spectral statistics in a disorder-free large-N setting and provide a symmetry-based roadmap for their random-matrix behavior across parameter space.

Abstract

We further explore the connection between holographic $O(n)$ tensor models and random matrices. First, we consider the simplest non-trivial uncolored tensor model and show that the results for the density of states, level spacing and spectral form factor are qualitatively identical to the colored case studied in arXiv:1612.06330. We also explain an overall 16-fold degeneracy by identifying various symmetries, some of which were unavailable in SYK and the colored models. Secondly, and perhaps more interestingly, we systematically identify the Spectral Mirror Symmetry and the Time-Reversal Symmetry of both the colored and uncolored models for all values of $n$, and use them to identify the Andreev ensembles that control their random matrix behavior. We find that the ensembles that arise exhibit a refined version of Bott periodicity in $n$.

Figures (7)

• Figure 1: MatrixPlot of the Hamiltonian \ref{['hamiltonian']}
• Figure 2: The density of states vs the energy levels. The spectrum has spectral mirror symmetry i.e., the spectrum is symmetric around $E=0$.
• Figure 3: The integrated density of states vs the energy levels
• Figure 4: SFF for $\beta =0$ for various window sizes ($\Delta t$). Similar to SYK and Gurau-Witten models, there is a dip-ramp-plateau structure present here as well, suggesting a random matrix/chaotic behaviour
• Figure 5: SFF for $\beta =0.05$ for various window sizes ($\Delta t$)