Topological expansion of the chain of matrices

Understand

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).