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- Title
Perturbing isoradial triangulations.
- Authors
David, François; Scott, Jeanne
- Abstract
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. 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 Δ(ϵ), the David-Eynard Kähler operator D(ϵ), and the conformal Laplacian Δ (ϵ) on the deformed Delaunay graph Gϵ. 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 Gcr. 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 Gϵ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.
- Subjects
GREEN'S functions; RANDOM graphs; CONFORMAL invariants; QUANTUM gravity; PLANAR graphs
- Publication
Annales de l'Institut Henri Poincaré D, 2024, Vol 11, Issue 4, p715
- ISSN
2308-5827
- Publication type
Article
- DOI
10.4171/AIHPD/178