The \(Z\)-invariant massive Laplacian on isoradial graphs
type: Item
instance of: scholarly article
title: The \(Z\)-invariant massive Laplacian on isoradial graphs
zbMATH Open document ID: 1372.82016
author: Cédric Boutillier
author: Béatrice de Tilière
author: Kilian Raschel
published in: Inventiones Mathematicae
publication date: 2017-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.00792
review text: A planar graph \(G=(V,E)\) (where \(V\) is the set of vertices and \(E\) is the set of edges) is called isoradial if it can be embedded in the plane in such a way that all internal faces are inscribable in a circle, with all circles having the same radius, and such that all circumcenters are in the interior of the faces. Assume that for each \(l=(xy)\in E\), a positive number (conductance) \(\rho(l)=\rho(xy)\) is given and that for each \(x\in V\), a mass \(m(x)\) is given. Then define the massive Laplacian operator \(\Delta^m:\mathbb C^V\to \mathbb C^V\) by \[ (\Delta^m)f(x)=d(x)f(x)-\sum_{y\sim x}\rho(xy)f(y), \] where \(d(x)=m^2(x)+ \sum_{y\sim x}\rho(xy)\). A function \(f\in \mathbb C^V\) is massive harmonic on \(G\) if \(\Delta^mf=0\). In the paper the authors introduce a one-parameter family of massive Laplacian operators \((\Delta^{m(k)})_{k\in [0,1)}\) in which \(\rho(xy)=\rho(\theta_{xy}|k)\) and \(m(x)=m(x|k)\) are given by \[ \rho(\theta_{xy}|k)=\text{sc}(\theta_{xy}|k), \;\;m^2(x|k)=\sum_{y\sim x}[A(\theta_{xy}|k)-\rho(\theta_{xy}|k)], \] where \(\text{sc}\) is one of Jacobi's twelve elliptic functions and the function \(A\) is related to integrals of squared Jacobi elliptic functions. An explicit formula for the inverse of \(\Delta^{m(k)}\), the massive Green function, is proved. It is shown that the Green function only depends on the local geometry of the graph and its asymptotics is computed. An explicit local formula for an infinite volume Boltzmann measure and for the free energy of the model are given. It is shown that the model has a second-order phase transition at \(k=2\). Moreover, it is proved that the massive Laplacian operator provides a one-parameter family of \(Z\)-invariant rooted spanning forest models. An explicit parametrization of the spectral curve of the massive Laplacian (when the graph is \(Z^2\)-periodic) is given. It is shown that the spectral curve is Harnack and has geometric genus 1.
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 6705730
zbMATH Keywords: massive Laplacian
zbMATH Keywords: isoradial graph
zbMATH Keywords: elliptic function
zbMATH Keywords: Green function
reviewed by: U. A. Rozikov
MaRDI profile type: Publication
OpenAlex ID: W2118020680
arXiv ID: 1504.00792
cites work: Classification of integrable equations on quad-graphs. The consistency approach
cites work: Q3347989
cites work: Q5558293
cites work: Q4793352
cites work: Q3999841
cites work: The critical \(Z\)-invariant Ising model via dimers: the periodic case
cites work: The critical \(Z\)-invariant Ising model via dimers: Locality property
cites work: A characterization of flip-accessibility for rhombus tilings of the whole plane
cites work: Uniform spanning forests
cites work: Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function
cites work: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances
cites work: Pseudoholomorphic simple Harnack curves
cites work: Q5302595
cites work: The critical temperature for the Ising model on planar doubly periodic graphs
cites work: Q5736202
cites work: LINE BUNDLES AND HOMOGENEOUS MATRICES
cites work: Q3905309
cites work: Q3905310
cites work: Quadri-tilings of the plane
cites work: Potential theory on a rhombic lattice
cites work: Q4293510
cites work: Tiling a polygon with parallelograms
cites work: The Laplacian and Dirac operators on critical planar graphs
cites work: Q4832435
cites work: Planar dimers and Harnack curves
cites work: Rhombic embeddings of planar quad-graphs
cites work: Elliptic functions and applications
cites work: Critical temperature of periodic Ising models
cites work: Phase transition free regions in the Ising model via the Kac-Ward operator
cites work: Discrete Riemann surfaces and the Ising model.
cites work: Exponentials Form a Basis of Discrete Holomorphic Functions on a Compact
cites work: Real algebraic curves, the moment map and amoebas
cites work: Amoebas of maximal area
cites work: Klein surfaces
cites work: Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
cites work: Analytic combinatorics in several variables.
cites work: Conway's Tiling Groups
cites work: Q4228482
DOI: 10.1007/S00222-016-0687-Z
description: scientific article