A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective spac
e. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $tau$-functions on the lattice.
Bilinear structure for the discrete Painleve I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painleve I equation is naturally
extended to a discrete coupled system. Corresponding matrix model is also mentioned.
