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

Bonzom, Valentin; Combes, Frédéric

Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 4, pp. 443-485.

A di�fficult 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. �rough Wick’s theorem, we show that the Feynman graph expansion of the expectation values of those polynomials enumerates meandric systems whose lower arch con�guration is obtained from the upper arch con�guration 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. �is 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 �rst time.

Classification : 05-XX, 00-XX
Mots-clés : Gaussian random tensors, regular edge-colored graphs, meanders, stabilized-interval-free permutations