The asymptotic determinant of the discrete Laplacian

Acta Mathematica - Tập 185 - Trang 239-286 - 2000
Richard Kenyon1
1CNRS UMR 8628 Laboratoire de Mathématique, Université de Paris-Sud, Orsay Cedex, France

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Ahlfors, L.,Complex Analysis, 3rd edition. McGraw-Hill, New York, 1978. Apostol, T.,Modular Functions and Dirichlet Series in Number Theory, 2nd edition. Graduate Texts in Math., 41. Springer-Verlag, New York, 1990. Burton, R. &Pemantle, R., Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances.Ann. Probab., 21 (1993), 1329–1371. Cohn, H., Kenyon, R. & Propp, J., A variational principle for domino tilings. To appear inJ. Amer. Math. Soc. Duplantier, B. &David, F., Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice.J. Statist. Phys., 51 (1988), 327–434. Destainville, N., Mosseri, R. &Bailly, F., Configurational entropy of codimension-one tilings and directed membranes.J. Statist. Phys., 87 (1997), 697–754. Guttmann, A. &Bursill, R., Critical exponent for the loop-erased self-avoiding walk by Monte-Carlo methods.J. Statist. Phys., 59 (1990), 1–9. Hayman, W. K.,Multivalent Functions. Cambridge Tracts in Math., 48. Cambridge Univ. Press, Cambridge, 1958. Kac, M., Can one hear the shape of a drum?Amer. Math. Monthly, 73:4, Part II (1966), 1–23. Kasteleyn, P. W., The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice.Physica, 27 (1961), 1209–1225. —, Graph theory and crystal physics, inGraph Theory and Theoretical Physics, pp. 43–110. Academic Press, London, 1967. Kenyon, R., Local statistics of lattice dimers.Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 591–618. Kenyon, R., Conformal invariance of domino tiling. To appear inAnn. Probab. Kesten, H., Relations between solutions to a discrete and continuous Dirichlet problem, inRandom Walks, Brownian Motion, and Interacting Particle Systems, pp. 309–321. Progr. Probab., 28. Birkhäuser Boston, Boston, MA, 1991. Kirchhoff, G., Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird.Ann. Phys. Chem., 72 (1847), 497–508. Kenyon, R., Propp, J. & Wilson, D., Trees and matchings.Electron. J. Combin., 7 (2000), Research Paper 25, 34 pp. (electronic). Lawler, G., A lower bound on the growth exponent for loop-erased random walk in two dimensions.ESAIM Probab. Statist., 3 (1999), 1–21 (electronic). Majumdar, S. N., Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions.Phys. Rev. Lett., 68 (1992), 2329–2331. McKean, H. &Singer, I., Curvature and the eigenvalues of the Laplacian.J. Differential Geom., 1 (1967), 43–69. McCoy, B. &Wu, T.,The Two-Dimensional Ising Model. Harvard Univ. Press, Cambridge, MA, 1973. Osgood, B., Phillips, R. &Sarnak, P., Extremals of determinants of Laplacians.J. Funct. Anal., 80 (1988), 148–211. Pemantle, R., Choosing a spanning tree for the integer lattice uniformly.Ann. Probab., 19 (1991), 1559–1574. Spitzer, F.,Principles of Random Walks, 2nd edition. Graduate Texts in Math., 34. Springer-Verlag, New York-Heidelberg, 1976. Temperley, H., Enumeration of graphs on a large periodic lattice, inCombinatorics (Aberystwyth, 1973), pp. 202–204. London Math. Soc. Lecture Note Ser., 13. Cambridge Univ. Press, London, 1974. Templerley, H. &Fisher, M., Dimer problem in statistical mechanics—an exact result.Philos. Mag. (8), 6 (1961), 1061–1063. Thurston, W. P., Conway's tiling groups.Amer. Math. Monthly, 97 (1990), 757–773.