All-orders asymptotics of tensor model observables from symmetries of restricted partitions

TL;DR

This paper derives the detailed large-degree asymptotic formulas for tensor model invariants, connecting combinatorial partition dominance with ribbon graph enumeration and algebraic structures, extending to general tensor ranks.

Contribution

It provides the first all-orders asymptotic expansion for the counting of tensor invariants and related combinatorial objects, revealing the role of partition dominance and symmetry factors.

Findings

Asymptotic expansion for $Z_3(n)$ derived and connected to ribbon graph enumeration.

Dominance of partitions with many parts of size 1 in asymptotics.

Conjectured formulas for invariants of general $d$-index tensors.

Abstract

The counting of the dimension of the space of $U(N) \times U(N) \times U(N)$ polynomial invariants of a complex $3$-index tensor as a function of degree $n$ is known in terms of a sum of squares of Kronecker coefficients. For $n \le N$, the formula can be expressed in terms of a sum of symmetry factors of partitions of $n$ denoted $Z_3(n)$. We derive the large $n$ all-orders asymptotic formula for $ Z_3(n)$ making contact with high order results previously obtained numerically. The derivation relies on the dominance in the sum, of partitions with many parts of length $1$. The dominance of other small parts in restricted partition sums leads to related asymptotic results. The result for the $3$-index tensor observables gives the large $n$ asymptotic expansion for the counting of bipartite ribbon graphs with $n$ edges, and for the dimension of the associated Kronecker permutation centralizer algebra. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The large $n$ dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general $d$-index tensors. The coefficients of $ 1/n$ in these expansions involve Stirling numbers of the second kind along with restricted partition sums.