Perturbing isoradial triangulations

Abstract

We consider an infinite planar Delaunay graph ${\mathbf{G}}_{ϵ}$ which is obtained by locally deforming the coordinate embedding of a general isoradial graph ${\mathbf{G}}_{\mathrm{cr}}$, with respect to a real deformation parameter $ϵ$. Using Kenyon’s exact and asymptotic results for the critical 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 Laplace–Beltrami operator $\Delta (ϵ)$, the David–Eynard Kähler operator $\mathcal{D}(ϵ)$, and the conformal Laplacian $\underline{\mathbf{\Delta }}(ϵ)$ on the deformed Delaunay graph ${\mathbf{G}}_{ϵ}$. We show that the scaling limits of the second-order bi-local term for both the Laplace–Beltrami and David–Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph ${\mathbf{G}}_{\mathrm{cr}}$. Our results allow us to define a discrete analog 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 Laplace–Beltrami and David–Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the 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 ${\mathbf{G}}_{ϵ}$; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.