Multi-critical tensor models and hard dimers on spherical random lattices

Random tensor models which display multi-critical behaviors in a remarkably simple fashion are presented. They come with entropy exponents γ=(m−1)/m$\gamma =(m-1)/m$, similarly to multi-critical random branched polymers. Moreover, they are interpreted as models of hard dimers on a set of random lattices for the sphere in dimension three and higher. Dimers with their exclusion rules are generated by the different interactions between tensors, whose coupling constants are dimer activities. As an illustration, we describe one multi-critical point, which is interpreted as a transition between the dilute phase and a crystallized phase, though with negative activities.