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We apply the general theory of Cauchy biorthogonal polynomials developedpreviously by the authors, to the case associated with Laguerre measures. Inparticular, we obtain explicit formulae in terms of Meijer-G functions for allkey objects relevant to the study of the corresponding biorthogonal polynomialsand the Cauchy two-matrix model associated with them. The central theorem weprove is that a scaling limit of the correlation functions for eigenvalues nearthe origin exists, and is given by a new determinantal two--level random pointfield, the "Meijer-G random field". We conjecture that this random point fieldleads to a novel universality class of random fields parametrized by exponentsof Laguerre weights. We express the joint distributions of the smallesteigenvalues in terms of suitable Fredholm determinants and evaluate themnumerically. We also show that in a suitable limit, the Meijer-G random fieldconverges to the Bessel random field and hence the behavior of the eigenvaluesof one of the two matrices converges to the one of the Laguerre ensemble.

Cauchy-Laguerre two-matrix model and the Meijer-G random point field