105 research outputs found
The expansion of tensor models with two symmetric tensors
It is well known that tensor models for a tensor with no symmetry admit a
expansion dominated by melonic graphs. This result relies crucially on
identifying \emph{jackets} which are globally defined ribbon graphs embedded in
the tensor graph. In contrast, no result of this kind has so far been
established for symmetric tensors because global jackets do not exist.
In this paper we introduce a new approach to the expansion in tensor
models adapted to symmetric tensors. In particular we do not use any global
structure like the jackets. We prove that, for any rank , a tensor model
with two symmetric tensors and interactions the complete graph admits
a expansion dominated by melonic graphs.Comment: misprints corrected, references adde
The prescription in the SYK model
We introduce an prescription for the SYK model both at
finite and at zero temperature. This prescription regularizes all the naive
ultraviolet divergences of the model. As expected the prescription breaks the
conformal invariance, but the latter is restored in the limit.
We prove rigorously that the Schwinger Dyson equation of the resummed two point
function at large and low momentum is recovered in this limit. Based on
this prescription we introduce an effective field theory
Lagrangian for the infrared SYK model.Comment: Second version: the effective field theory part of the paper
(subsections 2.1 and 3.1 and discussion) adde
Universality for Random Tensors
We prove two universality results for random tensors of arbitrary rank D. We
first prove that a random tensor whose entries are N^D independent, identically
distributed, complex random variables converges in distribution in the large N
limit to the same limit as the distributional limit of a Gaussian tensor model.
This generalizes the universality of random matrices to random tensors.
We then prove a second, stronger, universality result. Under the weaker
assumption that the joint probability distribution of tensor entries is
invariant, assuming that the cumulants of this invariant distribution are
uniformly bounded, we prove that in the large N limit the tensor again
converges in distribution to the distributional limit of a Gaussian tensor
model. We emphasize that the covariance of the large N Gaussian is not
universal, but depends strongly on the details of the joint distribution.Comment: Final versio
Regular colored graphs of positive degree
Regular colored graphs are dual representations of pure colored D-dimensional
complexes. These graphs can be classified with respect to an integer, their
degree, much like maps are characterized by the genus. We analyse the structure
of regular colored graphs of fixed positive degree and perform their exact and
asymptotic enumeration. In particular we show that the generating function of
the family of graphs of fixed degree is an algebraic series with a positive
radius of convergence, independant of the degree. We describe the singular
behavior of this series near its dominant singularity, and use the results to
establish the double scaling limit of colored tensor models.Comment: Final version. Significant improvements made, main results unchange
- …