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

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.