Universality of Heavy Operators in Matrix Models

TL;DR

The work analyzes universality of heavy operators in a simple bosonic two-matrix model (Hoppe model) at large $N$ and strong coupling, showing a universal black-hole–like regime where eigenvalue densities are parabolic and probe correlators depend on only a few parameters, underpinned by commuting matrices. It demonstrates that universal behavior persists across several heavy-operator families (potentials, classical sources, and characters) and extends to joint eigenvalue distributions via a commuting $X$ and $Y$ in the strong coupling limit, with a sharp phase boundary to a non-universal regime. The authors employ both analytic saddle-point analyses and Hybrid Monte Carlo simulations to map out universality, phase transitions, and an Abelianization mechanism wherein a single eigenvalue decouples and dominates heavy observables. They further discuss when Abelianization occurs, contrasting it with conventional saddles, and speculate on broader applicability to more complex matrix models and holographic contexts. Overall, the paper provides a concrete, tractable setting where black-hole–style universality arises in matrix models and outlines paths to generalize these ideas to richer theories of quantum gravity.

Abstract

In large $N$ theories with a gravity dual, generic heavy operators should be dual to black holes in the bulk. The microscopic details of such operators should then be irrelevant in the low energy theory. We look for such universality in the strong coupling limit of a very simple two matrix model -- the Hoppe model. Using analytics as well as Monte Carlo simulations, we show that there exists a universal black hole regime where the eigenvalue densities are given by parabolas and the correlation functions of probes in these backgrounds are completely determined by a few parameters. An important feature of strong coupling in this model is that the matrices commute and one can define joint eigenvalue distributions which also exhibit universality. These two results extend the beautiful findings of Berenstein, Hanada and Hartnoll. Not all heavy operators are universal and at strong coupling there is a sharp phase boundary between the universal and non universal regimes (Of course this should not be confused with the universality of eigenvalue spacing in matrix models). Moreover, in the non universal phase, we also find an interesting phenomenon we call Abelianization where some eigenvalues run off to infinity, reminiscent of heavy dual giant gravitons in $\mathcal N=4$ SYM.

Figures (15)

• Figure 1: A plot of moments for uniform line source $J$ of width $\alpha$. For $\alpha<\alpha_c$ all curves approach the same universal plateau. Highlighting this universality is the main purpose of this work; it shows up at strong coupling only. For $\alpha>\alpha_c$ this universality is gone. At large $\alpha$ a different, simpler, sort of universality emerges. That simpler universality of large sources is valid at any coupling.
• Figure 2: At zero coupling, $\lambda=0$, the densities interpolate smoothly between the semi-circle rule at vanishing source to a flat constant distribution for large source. (Here $N=300$ and the last point for each curve - where the density vanishes - was estimated through a fit.)
• Figure 3: Density $\rho(x)$ for the line source for various $\alpha$'s at strong coupling $\lambda=15000$. The densities are computed by solving the discretized version of \ref{['SPalpha']} with $N=1000$ points -- these are the solid lines on the left. On the right, we rescale the points to $\tilde{x}={x}/{\sqrt{\tfrac{1}{N}\langle{\rm\,tr\,} X^2\rangle}}$ such that they have unit variance. In the top row, we are in the universal regime $\alpha\leq \alpha_c\approx3.46$ and after rescaling, the densities collapse to the same curve! The densities also agree well with analytic parabolas \ref{['LengthAlpha']} shown as dashed red lines. Close to criticality, the agreement is not perfect due to finite coupling effects. In the bottom row, we are in the non-universal regime and the support of the densities is no longer $O(\lambda^{-\frac{1}{6}})$. We see here that the distribution is approaching $\rho(x)=1/\alpha$ at large $\alpha$.
• Figure 4: Eigenvalue density $\rho(x)$ for the quartic potential $V(x)=-\frac{x^2}{2}+g\frac{x^4}{4}$ at $g=\frac{1}{3}$ and $\lambda=15000$. We show here the leading saddle with has filling fractions $\mathcal{S}_1=\mathcal{S}_2=\frac{1}{2}$. The blue dots are from solving the discrete SPEs that follow from \ref{['spePotentialNoApprox']} with $N=1000$. They agree well with the dashed red lines which are the analytic parabolas \ref{['densityMultiCut']}.
• Figure 5: Top: Two simple examples of huge Character operators corresponding to a rectangular YT (in blue) and trapezium YT (in red). A rectangle with each row having $K$ boxes corresponds to a stack of $K$ determinants and a trapezium with $i^{\text{th}}$ row having $K+N-i$ boxes corresponds to a stack of determinants times a vandermonde-like factor (with sums instead of differences). At large $N$, we can define a density of the shifted highest weight $h_i$ and for the rectangle, it is a uniform distribution on the interval $\left[\tfrac{K}{N},\tfrac{K}{N}+1\right]$. For the trapezium, it is also a uniform distribution but on the interval $\left[\tfrac{K}{N},\tfrac{K}{N}+2\right]$. Bottom: Again, we find universal parabolas for these huge operators as described in the main text.