  For a given polynomial $V(x)\in \mathbb C[x]$, a random matrix eigenvalues
measure is a measure $\prod_{1\leq i<j\leq N}(x_i-x_j)^2 \prod_{i=1}^N
e^{-V(x_i)}dx_i$ on $\gamma^N$. Hermitian matrices have real eigenvalues
$\gamma=\mathbb R$, which generalize to $\gamma$ a complex Jordan arc, or
actually a linear combination of homotopy classes of Jordan arcs, chosen such
that integrals are absolutely convergent. Polynomial moments of such measure
satisfy a set of linear equations called "loop equations". We prove that every
solution of loop equations are necessarily polynomial moments of some random
matrix measure for some choice of arcs. There is an isomorphism between the
homology space of integrable arcs and the set of solutions of loop equations.
We also generalize this to a 2-matrix model and to the chain of matrices, and
to cases where $V$ is not a polynomial but $V'(x)\in \mathbb C(x)$.
