Wulff construction in statistical mechanics and in combinatorics

We present the geometric solutions to some variational problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.Comment: A revie

Random tensor models in the large N limit: Uncoloring the colored tensor models

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents \gamma_m=1-1/m.Comment: 15 page

The Multi-Orientable Random Tensor Model, a Review

After its introduction (initially within a group field theory framework) in [Tanasa A., J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694], the multi-orientable (MO) tensor model grew over the last years into a solid alternative of the celebrated colored (and colored-like) random tensor model. In this paper we review the most important results of the study of this MO model: the implementation of the $1/N$ expansion and of the large $N$ limit ($N$ being the size of the tensor), the combinatorial analysis of the various terms of this expansion and finally, the recent implementation of a double scaling limit