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.

arXiv comment44 pages

References in corpus (28)

• Gauge Theory Correlators from Non-Critical String Theory
• AdS3 Black Holes and a Stringy Exclusion Principle
• Invasion of the Giant Gravitons from Anti-de Sitter Space
• Critical behavior of colored tensor models in the large N limit
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• The complete 1/N expansion of colored tensor models in arbitrary dimension
• The Tensor Track, III
• Counting Tensor Model Observables and Branched Covers of the 2-Sphere
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• On vanishing of Kronecker coefficients
• Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics
• Schur-Weyl duality as an instrument of Gauge-String duality
• Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras
• Cut and join operator ring in Aristotelian tensor model
• Submatrix deconfinement and small black holes in AdS
• Nonvanishing of Kronecker coefficients for rectangular shapes
• Permutation Centralizer Algebras and Multi-Matrix Invariants
• Partition function of free conformal fields in 3-plet representation
• Orthogonal Bases of Invariants in Tensor Models
• On the counting of $O(N)$ tensor invariants
• Quantum error correction and large $N$
• Unitary matrix models and random partitions: Universality and multi-criticality
• From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$
• Generalized cut operation associated with higher order variation in tensor models
• Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
• On the Number of Conjugate Classes of Derangements
• Quiver Asymptotics: $\mathcal{N}=1$ Free Chiral Ring

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