Perturbing an isoradial triangulation

The theory of random Delaunay triangulations of the plane has been proposed by
David-Eynard and others as a discrete model for 2-dimensional quantum gravity: In this model
the role of a continuous metric is played by a Delaunay triangulation while flat metrics
correspond to isoradial triangulations (on which one can define a theory of discrete analyticity).
Like the continuous case, the partition function for this discrete theory is given by a suitably
normalized determinant of a Beltrami-Laplace operator which varies with the choice of
triangulation. An elegant formula of Richard Kenyon expresses this determinant as a finite sum
of local contributions when the triangulation is both isoradial and periodic. In joint work with
François David we consider smooth perturbations of a periodic isoradial triangulation and obtain
an asymptotic expansion for the second variation of the log-determinant of the discrete
Beltrami-Laplace operator. This result can be interpreted as a discretization of the formula for
the second variation (of the logarithm) of the continuous partition function know from conformal
field theory; using this interpretation we can identify the central charge in this discrete setting.