An analysis of the intermediate field theory of $T^4$ tensor model

In this paper we analyze the multi-matrix model arising from the intermediate field representation of the tensor model with all quartic melonic interactions. We derive the saddle point equation and the Schwinger-Dyson constraints. We then use them to describe the leading and next-to-leading eigenvalues distribution of the matrices.

The sine-law gap probability, Painleve 5, and asymptotic expansion by the topological recursion

The goal of this article is to rederive the connection between the Painleve $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, textbf{to all orders}, that the WKB asymptotic expansions of the $tau$-function as well as of determinantal formulas arising from the Painleve $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the sine-law universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.

Abstract loop equations, topological recursion, and applications

We formulate a notion of abstract loop equations, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture

The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been verified to low genus for several toric CY3folds, and proved to all genus only for C^3. In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model.Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in 2 steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kahler radius coincide due to special geometry property implied by the topological recursion.

Universal scaling limits of matrix models, and (p,q) Liouville gravity

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ~ x^{p/q}, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p,q)-minimal models kernels. Those (p,q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree at most p. For example, near a regular edge y ~ x^{1/2}, the (1,2) kernel is the Airy kernel and we recover the Airy law. Those kernels are associated to the (p,q) minimal model, i.e. the (p,q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which q=2.

Topological expansion of the chain of matrices

We solve the loop equations to all orders in $1/N^2$, for the Chain of Matrices matrix model (with possibly an external field coupled to the last matrix of the chain). We show that the topological expansion of the free energy, is, like for the 1 and 2-matrix model, given by the symplectic invariants of the associated spectral curve. As a consequence, we find the double scaling limit explicitly, and we discuss modular properties, large $N$ asymptotics. We also briefly discuss the limit of an infinite chain of matrices (matrix quantum mechanics).