Communications in Mathematical Physics

, Volume 330, Issue 3, pp 973–1019 | Cite as

The 1/N Expansion of Tensor Models Beyond Perturbation Theory



We analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor. Using a version of the Loop Vertex Expansion (which we call the mixed expansion) we show that the cumulants write as explicit series in 1/N plus bounded rest terms. The mixed expansion recasts the problem of determining the subleading corrections in 1/N into a simple combinatorial problem of counting trees decorated by a finite number of loop edges.

As an aside, we use the mixed expansion to show that the (divergent) perturbative expansion of the tensor models is Borel summable and to prove that the cumulants respect an uniform scaling bound. In particular the quartically perturbed measures fall, in the N→ ∞ limit, in the universality class of Gaussian tensor models.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CPHT-UMR 7644, CNRSÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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