The calculation of expectation values in Gaussian random tensor theory via meanders

Abstract

A difficult problem in the theory of random tensors is to calculate the expectation values of polynomials in the tensor entries, even in the large N limit and in a Gaussian distribution. Here we address this challenge for tensors of rank 4, focusing on a family of polynomials labeled by permutations, which generalize in a precise sense the single-trace invariants of random matrix models. Through Wick’s theorem, we show that the Feynman graph expansion of the expectation values of those polynomials enumerates meandric systems whose lower arch configuration is obtained from the upper arch configuration by a permutation on half of the arch feet. Our main theorem reduces the calculation of expectation values to those of polynomials labeled by stabilized-interval-free permutations (SIF) which are proved to enumerate irreducible meandric systems. This together with explicit calculations of expectation values associated to SIF permutations allows to exactly evaluate large N expectation values beyond the so-called melonic polynomials for the first time.