Abstract:Année de publication : 2021
We consider an infinite, planar, Delaunay graph Gε which is obtained by locally deforming the coordinate embedding of a general, isoradial graph Gcr, with respect to a real deformation parameter ε. This entails a careful analysis of Whitehead edge-flips induced by the deformation and the Delaunay constraints. Using R. Kenyon’s exact and asymptotic results for the Green’s function on an isoradial graph, we calculate the leading asymptotics of the first and second order terms in the perturbative expansion of the log- determinant of the Beltrami-Laplace operator ∆(ε), the David-Eynard Kähler operator D(ε), and the conformal Laplacian ∆(ε) on the deformed Delaunay graph Gε. We show that the scaling limits of the second order bi-local term for both the Beltrami-Laplace and David-Eynard Kähler operators exist and co- incide, with a shared value independent of the choice of initial isoradial graph Gcr. Our results allow us to define a discrete analogue of the stress energy tensor for each of the three operators. Furthermore we can identify a central charge (c = −2) in the case of both the Beltrami-Laplace and David-Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c = −2 for the David-Eynard Kähler operator is, however, at odds with the value of c = −26 expected by Polyakov’s theory of 2D quantum gravity; moreover there are problems with convergence of the scaling limit of the discrete stress energy tensor for the David-Eynard Kähler operator. The second order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph Gε; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.