  We apply the general theory of Cauchy biorthogonal polynomials developed
previously by the authors, to the case associated with Laguerre measures. In
particular, we obtain explicit formulae in terms of Meijer-G functions for all
key objects relevant to the study of the corresponding biorthogonal polynomials
and the Cauchy two-matrix model associated with them. The central theorem we
prove is that a scaling limit of the correlation functions for eigenvalues near
the origin exists, and is given by a new determinantal two--level random point
field, the "Meijer-G random field". We conjecture that this random point field
leads to a novel universality class of random fields parametrized by exponents
of Laguerre weights. We express the joint distributions of the smallest
eigenvalues in terms of suitable Fredholm determinants and evaluate them
numerically. We also show that in a suitable limit, the Meijer-G random field
converges to the Bessel random field and hence the behavior of the eigenvalues
of one of the two matrices converges to the one of the Laguerre ensemble.
