Tue 14 Dec 2021, 11:00 (Paris time)
Jeanne Scott
Perturbing Isoradial Triangulations
(Slides)
Location: Sorbonne Université, Jussieu, room 16-26-113
In joint work with François David, we consider an infinite, planar Delaunay
graph $\mathcal{G}_\epsilon$ which is obtained by locally perturbing the
coordinate embedding of a general, isoradial graph $\mathcal{G}_\mathrm{cr} $
with respect to real-valued, deformation parameter(s) $\epsilon$.
Three Laplace-like operators are supported on such a Delaunay graph
$\mathcal{G}_\epsilon$, namely: The Beltrami-Laplace operator
$\Delta(\epsilon)$, the David-Eynard Kähler operator $\mathcal{D}(\epsilon)$,
and the conformal Laplacian $\underline{\Delta}(\epsilon)$.
All three operators coincide with
Kenyon's critical Laplacian when restricted to the initial, isoradial graph
$\mathcal{G}_\mathrm{cr}$ and, as such, each operator can be viewed as a
independent deformation of the critical Laplacian. Using Kenyon's asymptotic
results for the Green's function of the critical Laplacian, we calculate the
leading asymptotics of the first and second order terms in the perturbative
expansion of the log-determinant of each operator. The first order term allow
us to define a discrete stress energy tensor for each operator. In the case of
the Beltrami-Laplace and discrete Kähler operators, the second order term share
a convergent scaling limit which is independent of the initial isoradial graph
$\mathcal{G}_\epsilon$; for the Beltrami-Laplace operator this limit is
consistent with the results of the continuum theory (i.e. the GFF).