Tensor Models and Random Geometry

Invariant rank-$d$ tensor integrals generate colored $d$-dimensional cellular complexes as Feynman graphs. Their large-$N$ expansion is controlled by the Gurau degree, with melonic graphs dominant in standard scalings. This gives calculability and critical limits, but the leading geometry is generally branched-polymer-like rather than a smooth Einstein spacetime.

Required background. Matrix Eigenvalue Saddles, Loop Equations, and Phase Transitions supplies the rank-two comparison.

Helpful background. Euclidean Dynamical Triangulations and Their Relation to CDT, From Genus Counting to a Holographic String Regime, and Tensor Renormalization of Euclidean Path Integrals supply geometric and RG comparisons.

Invariant tensor integrals

Let $T_{a_1\ldots a_d}$ transform under $U(N)^d$. A quartic melonic model is

$$Z=\int dT\,d\bar T\, \exp\left[-N^{d-1} \left(\bar T\cdot T+\lambda\sum_{c=1}^{d}V_c[T,\bar T]\right)\right],$$

where $V_c$ contracts all indices pairwise except for a color-$c$ exchange. Propagator strands and colored interaction vertices define a dual cellular complex.

For a connected vacuum graph $\mathcal G$, the amplitude has the form

$$\mathcal A_{\mathcal G}\propto N^{d-\frac{2}{(d-1)!}\omega(\mathcal G)},$$

where the nonnegative Gurau degree $\omega$ generalizes matrix genus. Degree-zero graphs dominate Gurau 2011.

Application: the melonic two-point equation

Normalize the full large-$N$ two-point function to $G(0)=1$. For a simple quartic melonic convention, recursive insertion gives

$$G=1+qG^2,\qquad q=d\lambda .$$

Choosing the branch analytic at $q=0$,

$$G(q)=\frac{1-\sqrt{1-4q}}{2q}.$$

The critical point is $q_c=1/4$, with square-root singularity. Iterating the self-energy inserts melons on propagator lines, proving their dominance and producing a controlled critical graph ensemble.

This calculation establishes large-$N$ counting and criticality. It does not establish a continuum metric with four-dimensional local graviton dynamics. Standard melonic ensembles share branched-polymer critical behavior, a major limitation for geometric interpretation.

Geometry and universality tests

Measure graph distance, Hausdorff and spectral dimensions, topology proxies, correlation length, and matter observables while approaching criticality. Then vary interaction invariants and rank. A continuum claim requires stable renormalized observables, not only divergence of graph number.

Double-scaling limits retain subleading degrees by correlating $N\to\infty$ with $q\to q_c$. Whether that enlarged graph family changes the continuum geometry is an observable question; subleading topology alone does not guarantee Einstein behavior.

Adversarial dominance test

Add a nonmelonic interaction with a scaling that competes at large $N$. Recompute the Schwinger–Dyson equation and critical exponents. If the proposed continuum data change, the earlier result belongs to a specific universality class. Also compare at fixed graph volume to remove critical-size bias.

Scope

Tensor models provide genuine higher-rank expansions and exact random-geometry results. Their connection to spacetime requires additional continuum, causality, and classical-recovery evidence. Matrix-Model Lessons Beyond Two Dimensions makes the analogy precise.

The chapter overview compares these assumptions and limitations.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bonzom, V., R. Gurau, A. Riello, and V. Rivasseau. “Critical Behavior of Colored Tensor Models in the Large N Limit.” Nuclear Physics B 853 (2011): 174–195. DOI.
• Gurau, R. “The 1/N Expansion of Colored Tensor Models.” Annales Henri Poincaré 12 (2011): 829–847. DOI.