Twofold universality of large-$N$ melonic random tensors

We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors $T_{a_1,a_2,\ldots,a_D} \in \mathbb{C}$ ($a_1,\ldots,a_D=1,\ldots,N$) at large $N$.

In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a $D$-coloured graph $B$) to contract their indices and form a tensor trace $B(T)$.

We show that, whenever each $B_1,\ldots, B_n$ can be obtained by a recursive construction known as melonicity, then the leading order in $N$ of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the -- often intricate -- combinatorics of the traces $B_i$, but also, to our surprise, independent of $D$ as far as $D\geq 3$.

Instead, at large $N$, these integrals are some functions (indexed by $n$) of the number of vertices $2p_i$ of $B_i$ which we call melonic polynomials.

Melonic traces cumulants with respect to any ('interacting') measure \[ \exp\Big\{-N^{D-1} \sum_{i=1}^m g_i {B_i}(T)\Big\} \mathrm{d}\mu_0(T) \quad (g_1,\ldots,g_m \in \mathbb{R}, \mathrm{d}\mu_0(T) =\text{the tensor Gaussian}) \] with each $B_i$ melonic, can be computed with our universal measure that replaces each $B_i$ by a canonical trace depending only on $p_i$.

We prove that any two melonic tensor models are indistinguishable at large-$N$, independently of the number of tensor indices (first universality aspect), and of the fine-grainedness of their interactions (second universality), being a sufficient condition that the couplings (the parameters $g_i$ above) agree and their respective traces are monomials with the same degree in $T$.