Perturbing Isoradial Triangulations

TL;DR

This paper analyzes how small deformations of isoradial graphs affect associated Laplacian operators, revealing asymptotic behaviors, defining discrete stress tensors, and exploring connections to conformal field theory and statistical models.

Contribution

It provides a detailed asymptotic analysis of Laplacian determinants on deformed isoradial graphs and introduces a discrete stress energy tensor concept.

Findings

Second order bi-local terms converge and are graph-independent.

Discrete central charge c=-2 for David-Eynard operator contrasts with theoretical expectations.

Scaling limits for the stress energy tensor show partial consistency with Gaussian free field.

Abstract

We consider an infinite, planar, Delaunay graph which is obtained by locally deforming the embedding of a general, isoradial graph, w.r.t. a real deformation parameter $\epsilon$. This entails a careful analysis of edge-flips induced by the deformation and the Delaunay constraints. Using 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 $\Delta(\epsilon)$, the David-Eynard K\"ahler operator $\mathcal{D}(\epsilon)$, and the conformal Laplacian $\underline{\Delta}(\epsilon)$ on the deformed graph. We show that the scaling limits of the second order {\it bi-local} term for both the Beltrami-Laplace and David-Eynard operators exist and coincide, with a value independent of the choice of initial isoradial graph. Our results allow to define a discrete analogue of the stress energy tensor for each of the three operators. Furthermore we can identify a central charge ($c$) in the case of both the Beltrami-Laplace and David-Eynard 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 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 operator. The bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete {\it curvature dipoles} in the deformed Delaunay graph; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.