Comptes Rendus
no. 6
Probabilité, Géométrie différentielle, Analyse globale sur les variétés
Determinants of Laplacians on discretizations of flat surfaces and analytic torsion
[Déterminants de laplaciens sur les discrétisations de surfaces plates et torsion analytique.]
Siarhei Finski1
1 Institut Fourier - Université Grenoble Alpes, France
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 743-751.

Nous étudions le développement asymptotique des déterminants de laplaciens des graphes associés aux discrétisations d’une surface de demi-translation munie d’un fibré vectoriel hermitien plat. Ainsi, sur les discrétisations, nous mettons en relation les déterminants zêta-régularisés avec le développement asymptotique du nombre d’arbres couvrants et la fonction de partition des forêts couvrantes d’unicycles d’un poids déterminé par la monodromie de la connexion unitaire du fibré vectoriel hermitien.

We study the asymptotic expansion of the determinants of the graph Laplacians associated to discretizations of a half-translation surface endowed with a unitary flat vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests, weighted by the monodromy of the unitary connection of the vector bundle, to the corresponding zeta-regularized determinants.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.94
Siarhei Finski 1

1 Institut Fourier - Université Grenoble Alpes, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Siarhei Finski. Determinants of Laplacians on discretizations of flat surfaces and analytic torsion. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 743-751. doi : 10.5802/crmath.94. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.94/

[1] Robert A. Adams Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | MR | Zbl

[2] Clara L. Aldana; Julie Rowlett A Polyakov formula for sectors, J. Geom. Anal., Volume 28 (2018) no. 2, pp. 1773-1839 | DOI | MR | Zbl

[3] Jeff Cheeger Spectral geometry of singular Riemannian spaces, J. Differ. Geom., Volume 18 (1983) no. 4, pp. 575-657 | DOI | MR | Zbl

[4] Jean-Pierre Demailly Complex Analytic and Differential Geometry, 2012 www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[5] Julien Dubédat Dimers and families of Cauchy-Riemann operators. I, J. Am. Math. Soc., Volume 28 (2015) no. 4, pp. 1063-1167 | DOI | MR | Zbl

[6] Julien Dubédat; Reza Gheissari Asymptotics of height change on toroidal Temperleyan dimer models, J. Stat. Phys., Volume 159 (2015) no. 1, pp. 75-100 | DOI | MR | Zbl

[7] Richard J. Duffin Discrete potential theory, Duke Math. J., Volume 20 (1953), pp. 233-252 | DOI | MR | Zbl

[8] Bertrand Duplantier; François David Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, J. Stat. Phys., Volume 51 (1988) no. 3-4, pp. 327-434 | DOI | MR | Zbl

[9] Siarhei Finski Finite-difference method on flat surfaces with a unitary flat vector bundle (2020) | arXiv

[10] Siarhei Finski Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion (2020) | arXiv

[11] Robin Forman Determinants of Laplacians on graphs, Topology, Volume 32 (1993) no. 1, pp. 35-46 | DOI | MR | Zbl

[12] Pierre Grisvard Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman Advanced Publishing Program, 1985 | MR | Zbl

[13] Adrien Kassel; Richard Kenyon Random curves on surfaces induced from the Laplacian determinant, Ann. Probab., Volume 45 (2017) no. 2, pp. 932-964 | DOI | MR | Zbl

[14] Richard Kenyon The asymptotic determinant of the discrete Laplacian, Acta Math., Volume 185 (2000) no. 2, pp. 239-286 | DOI | MR | Zbl

[15] Richard Kenyon The Laplacian and Dirac operators on critical planar graphs, Invent. Math., Volume 150 (2002) no. 2, pp. 409-439 | DOI | MR | Zbl

[16] Richard Kenyon Spanning forests and the vector bundle Laplacian, Ann. Probab., Volume 39 (2011) no. 5, pp. 1983-2017 | DOI | MR | Zbl

[17] Edith A. Mooers Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math., Volume 78 (1999), pp. 1-36 | DOI | MR | Zbl

[18] Werner Müller Analytic torsion and R-torsion of Riemannian manifolds., Adv. Math., Volume 28 (1978), pp. 233-305 | DOI | MR | Zbl

[19] Brad Osgood; Ralph Phillips; Peter Sarnak Extremals of determinants of Laplacians, J. Funct. Anal., Volume 80 (1988) no. 1, pp. 148-211 | DOI | MR | Zbl

[20] Daniel B. Ray; Isadore M. Singer Analytic torsion, Partial differential equations (Derkeley, 1971) (Proceedings of Symposia in Pure Mathematics), Volume 23, American Mathematical Society, 1973, pp. 167-181 | Zbl

[21] Daniel B. Ray; Isadore M. Singer Analytic torsion for complex manifolds, Ann. Math., Volume 98 (1973) no. 1, pp. 154-177 | MR | Zbl

[22] Michael Reed; Barry Simon Methods of modern mathematical physics. II: Fourier analysis, self-adjointness, Academic Press Inc., 1975 | Zbl

[23] Anton Zorich Flat surfaces, Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems (Papers from the meeting, Les Houches, France, March 9–21, 2003), Springer, 2006, pp. 437-583 | Zbl

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