Towards a quantitative characterization of gravitational universality classes for order-4 random tensor models

TL;DR

This paper investigates whether order-4 random tensor models can realize a continuum limit analogous to the Reuter fixed point in quantum gravity by applying a functional renormalization group (FRG) with tensor-size $N$ acting as the RG scale. Using an $O(N)^{\otimes 4}$-symmetric melonic truncation up to $T^8$, and a two-parameter regulator $R_N^{(\alpha,r)}$, the authors map fixed points across regulator space and identify three candidates, A, B, and C, with A not remaining real and B potentially compatible with Reuter only if real. They find that A has two relevant directions and is unlikely to match Reuter, while B can have three if real but is regulator-dependent; C remains robust with two relevant directions, suggesting a distinct universality class. The study emphasizes the role of regulator optimization (minimal sensitivity) and asymptotic limits ($\alpha\to0$, $\alpha\to\infty$) in assessing universality, and concludes that simple melonic tensor models likely do not realize the Reuter fixed point, motivating future work to include richer couplings and causality-inspired constraints. Overall, the work provides evidence for multiple, regulator-sensitive universality classes in tensor models and highlights the challenges of matching continuum quantum gravity fixed points within these discrete frameworks.

Abstract

Random tensor models can be used as combinatorial devices to generate Euclidean dynamical triangulations. A physical continuum limit of dynamical triangulations requires a suitable generalization of the double-scaling limit of random matrices. This limit corresponds to a fixed point of a pregeometric Renormalization Group flow in which the tensor size $N$ serves as the Renormalization Group scale. We search for corresponding fixed points in order-4 random tensor models associated to dynamical triangulations in 4 dimensions. In a $O(N)^{\otimes 4}$ symmetric setting, we discuss the resulting phase portrait as a function of the regulator parameters. We optimize our results, identifying parameter values for which the results are minimally sensitive to parameter changes. We find three fixed-point candidates: only one of them is real across the entire parameter range, but only has two relevant directions. This should be contrasted with the university class of the Reuter fixed point in continuum quantum gravity, very likely characterized by three relevant directions. We conclude that simple combinatorial models of Euclidean triangulations and the Reuter fixed point most likely lie in different universality classes.

Figures (10)

• Figure 1: Logarithmic plot of the regulator \ref{['eq:Regulator']} for different parameter values $(\alpha,r)$. The $x$-axis corresponds to $\frac{a+b+c+d}{N}$. Changes in $\alpha$ correspond to an overall rescaling without changing the shape of the function; changes in $r$ alter the shape.
• Figure 2: Real part of the critical exponents for the $T^4,T^6$ and $T^8$ truncations $(\alpha=1, r=1)$. With respect to Eichhorn:2019hsa we find two new fixed points, denoted $T_B^8$ and $T_C^8$ with roughly similar sets of critical exponents. The critical exponents with large degeneracy (i.e. the horizontal sequences of points in the figure corresponding roughly to $i=7,8\dots 20$ ) are all associated to couplings that vanish at the fixed point: the corresponding operators appear to receive a universal "dressing" of their critical exponents by the non-zero interactions.
• Figure 3: Plot of the real parts of the critical exponents for fixed-point candidate A in the top row, and fixed-point candidate B in the bottom row. The line $\alpha_{\rm crit}(r)$ corresponds to a smooth fit to the fixed point collision and is shown in magenta.
• Figure 4: Upper panels: Real part of the first three critical exponents for fixed points A and B in the eight-order truncation for $\alpha=1$. The dots correspond to the data points obtained by solving the $\beta$-functions on a grid stepsize of $\delta r=0.05$. The lines correspond to polynomial fits of order $4$ to $12$ used to obtain the minimal sensitivity $r$-values shown in Tab. \ref{['tab:a1_MS_fits']}. In panel (a), the blue and red vertical lines indicate the values of $r$ corresponding to the minimal sensitivity of the critical exponents $\theta_{1,2}$ of fixed points A and B, respectively. In panel (b), the blue and red line, indicating the value of $r$ selected by the principle of minimal sensitivity, agrees for both fixed points. Lower panels: Real part of the first three critical exponents for fixed points A and B in the eight-order truncation for $r=1$. The dots correspond to the data points obtained by solving the $\beta$-functions on a grid step size of $\delta \alpha=0.25$. The lines correspond to polynomial fits of order $4$ to $12$ used to obtain the minimal sensitivity $\alpha$-values. These are shown in Tab. \ref{['tab:r1_MS_fits']}. (a) The blue and red vertical lines correspond to the values of $r$ corresponding to the minimal sensitivity of the critical exponents $\theta_{1,2}$ of fixed points A and B, respectively. (b) The blue and red vertical lines correspond to the values of $r$ corresponding to the minimal sensitivity of the critical exponent $\theta_3$ of fixed points A and B, respectively.
• Figure 5: Critical exponents for the fixed point C in the eight-order truncation.