Dimers and families of Cauchy-Riemann operators I

Dubédat, Julien

doi:10.1090/jams/824

Dimers and families of Cauchy-Riemann operators I
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by Julien Dubédat

J. Amer. Math. Soc. 28 (2015), 1063-1167

DOI: https://doi.org/10.1090/jams/824

Published electronically: April 6, 2015

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Abstract:

In the dimer model, a configuration consists of a perfect matching of a fixed graph. If the underlying graph is planar and bipartite, such a configuration is associated to a height function. For appropriate “critical” (weighted) graphs, this height function is known to converge in the fine mesh limit to a Gaussian free field, following in particular Kenyon’s work.

In the present article, we study the asymptotics of smoothed and local field observables from the point of view of families of Cauchy-Riemann operators and their determinants. This allows one in particular to obtain a functional invariance principle for the field; characterise completely the limiting field on toroidal graphs as a compactified free field; analyze electric correlators; and settle the Fisher-Stephenson conjecture on monomer correlators.

The analysis is based on comparing the variation of determinants of families of (continuous) Cauchy-Riemann operators with that of their discrete (finite dimensional) approximations. This relies in turn on estimating precisely inverting kernels, in particular near singularities. In order to treat correlators of “singular” local operators, elements of (multiplicatively) multivalued discrete holomorphic functions are discussed.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Luis Alvarez-Gaumé, Jean-Benoît Bost, Gregory Moore, Philip Nelson, and Cumrun Vafa, Bosonization on higher genus Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 3, 503–552. MR 908551, DOI 10.1007/BF01218489
Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300, DOI 10.1214/EJP.v6-96
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. MR 2273508
Alexei Borodin and Patrik L. Ferrari, Anisotropic growth of random surfaces in $2+1$ dimensions, Comm. Math. Phys. 325 (2014), no. 2, 603–684. MR 3148098, DOI 10.1007/s00220-013-1823-x
Cédric Boutillier and Béatrice de Tilière, Loop statistics in the toroidal honeycomb dimer model, Ann. Probab. 37 (2009), no. 5, 1747–1777. MR 2561433, DOI 10.1214/09-AOP453
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396, DOI 10.1007/978-3-642-52244-4
Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs, Adv. Math. 228 (2011), no. 3, 1590–1630. MR 2824564, DOI 10.1016/j.aim.2011.06.025
David Cimasoni and Nicolai Reshetikhin, Dimers on surface graphs and spin structures. I, Comm. Math. Phys. 275 (2007), no. 1, 187–208. MR 2335773, DOI 10.1007/s00220-007-0302-7
Mihai Ciucu, Dimer packings with gaps and electrostatics, Proc. Natl. Acad. Sci. USA 105 (2008), no. 8, 2766–2772. MR 2383565, DOI 10.1073/pnas.0710659105
Mihai Ciucu, The emergence of the electrostatic field as a Feynman sum in random tilings with holes, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4921–4954. MR 2645056, DOI 10.1090/S0002-9947-10-05087-7
Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), no. 2, 297–346. MR 1815214, DOI 10.1090/S0894-0347-00-00355-6
Béatrice de Tilière, Quadri-tilings of the plane, Probab. Theory Related Fields 137 (2007), no. 3-4, 487–518. MR 2278466, DOI 10.1007/s00440-006-0002-9
B. de Tilière, Scaling limit of isoradial dimer models and the case of triangular quadri-tilings, Ann. Inst. H. Poincaré Probab. Stat. 43 (2007), no. 6, 729–750., DOI 10.1016/j.anihpb.2006.10.002
Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR 1424041, DOI 10.1007/978-1-4612-2256-9
P. Di Francesco, H. Saleur, and J.-B. Zuber, Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models, J. Statist. Phys. 49 (1987), no. 1-2, 57–79. MR 923852, DOI 10.1007/BF01009954
Julien Dubédat, SLE and the free field: partition functions and couplings, J. Amer. Math. Soc. 22 (2009), no. 4, 995–1054. MR 2525778, DOI 10.1090/S0894-0347-09-00636-5
Julien Dubédat, Topics on abelian spin models and related problems, Probab. Surv. 8 (2011), 374–402. MR 2861134, DOI 10.1214/11-PS187
R. J. Duffin, Potential theory on a rhombic lattice, J. Combinatorial Theory 5 (1968), 258–272. MR 232005, DOI 10.1016/S0021-9800(68)80072-9
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
John Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc. 96 (1992), no. 464, vi+123. MR 1146600, DOI 10.1090/memo/0464
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
Michael E. Fisher and John Stephenson, Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers, Phys. Rev. (2) 132 (1963), 1411–1431. MR 158705, DOI 10.1103/PhysRev.132.1411
Krzysztof Gawȩdzki, Lectures on conformal field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 727–805. MR 1701610, DOI 10.1007/s002200050573
James Glimm and Arthur Jaffe, Quantum physics, 2nd ed., Springer-Verlag, New York, 1987. A functional integral point of view. MR 887102, DOI 10.1007/978-1-4612-4728-9
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
Geoffrey R. Grimmett and Ioan Manolescu, Inhomogeneous bond percolation on square, triangular and hexagonal lattices, Ann. Probab. 41 (2013), no. 4, 2990–3025. MR 3112936, DOI 10.1214/11-AOP729
Robert E. Hartwig, Monomer pair correlations, J. Mathematical Phys. 7 (1966), 286–299. MR 187830, DOI 10.1063/1.1704931
Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
Nam-Gyu Kang and Nikolai G. Makarov, Gaussian free field and conformal field theory, Astérisque 353 (2013), viii+136 (English, with English and French summaries). MR 3052311
P. W. Kasteleyn, The statistics of dimers on a lattice. i. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209–1225.
Richard Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 5, 591–618 (English, with English and French summaries). MR 1473567, DOI 10.1016/S0246-0203(97)80106-9
Richard Kenyon, The asymptotic determinant of the discrete Laplacian, Acta Math. 185 (2000), no. 2, 239–286. MR 1819995, DOI 10.1007/BF02392811
Richard Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, 759–795. MR 1782431, DOI 10.1214/aop/1019160260
Richard Kenyon, Dominos and the Gaussian free field, Ann. Probab. 29 (2001), no. 3, 1128–1137. MR 1872739, DOI 10.1214/aop/1015345599
R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2002), no. 2, 409–439. MR 1933589, DOI 10.1007/s00222-002-0249-4
Richard Kenyon, Height fluctuations in the honeycomb dimer model, Comm. Math. Phys. 281 (2008), no. 3, 675–709. MR 2415464, DOI 10.1007/s00220-008-0511-8
Richard Kenyon, Lectures on dimers, Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., Providence, RI, 2009, pp. 191–230. MR 2523460, DOI 10.1090/pcms/016/04
Richard Kenyon, Conformal invariance of loops in the double-dimer model, Comm. Math. Phys. 326 (2014), no. 2, 477–497. MR 3165463, DOI 10.1007/s00220-013-1881-0
Richard Kenyon, Andrei Okounkov, and Scott Sheffield, Dimers and amoebae, Ann. of Math. (2) 163 (2006), no. 3, 1019–1056. MR 2215138, DOI 10.4007/annals.2006.163.1019
Richard Kenyon and Jean-Marc Schlenker, Rhombic embeddings of planar quad-graphs, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3443–3458 (English, with English and French summaries). MR 2146632, DOI 10.1090/S0002-9947-04-03545-7
Richard W. Kenyon, James G. Propp, and David B. Wilson, Trees and matchings, Electron. J. Combin. 7 (2000), Research Paper 25, 34. MR 1756162, DOI 10.37236/1503
Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
Christian Mercat, Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), no. 1, 177–216. MR 1824204, DOI 10.1007/s002200000348
C. Mercat, Discrete Polynomials and Discrete Holomorphic Approximation (2002). ArXiv Mathematical Physics e-prints.
Bernard Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys. 34 (1984), no. 5-6, 731–761. MR 751711, DOI 10.1007/BF01009437
B. Nienhuis and H. J. F. Knops, Spinor exponents for the two-dimensional potts model, Phys. Rev. B 32 (1985), 1872–1875., DOI 10.1103/PhysRevB.32.1872
John Palmer, Determinants of Cauchy-Riemann operators as $\tau$-functions, Acta Appl. Math. 18 (1990), no. 3, 199–223. MR 1065639, DOI 10.1007/BF00049126
Jerome K. Percus, One more technique for the dimer problem, J. Mathematical Phys. 10 (1969), 1881–1888. MR 250899, DOI 10.1063/1.1664774
Haru Pinson, Rotational invariance and discrete analyticity in the 2d dimer model, Comm. Math. Phys. 245 (2004), no. 2, 355–382. MR 2039700, DOI 10.1007/s00220-003-1021-3
V. B. Priezzhev and P. Ruelle, Boundary monomers in the dimer model, Phys. Rev. E 77 (2008), 061126., DOI 10.1103/PhysRevE.77.061126
D. Kvillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 37–41, 96 (Russian). MR 783704
D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI 10.2307/1970909
Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521–541. MR 2322706, DOI 10.1007/s00440-006-0050-1
Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974. MR 0489552
Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
Stanislav Smirnov, Towards conformal invariance of 2D lattice models, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421–1451. MR 2275653
Stanislav Smirnov, Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. of Math. (2) 172 (2010), no. 2, 1435–1467. MR 2680496, DOI 10.4007/annals.2010.172.1441
Daniel W. Stroock, Partial differential equations for probabilists, Cambridge Studies in Advanced Mathematics, vol. 112, Cambridge University Press, Cambridge, 2012. Paperback edition of the 2008 original. MR 2932517
H. van Beijeren, Exactly solvable model for the roughening transition of a crystal surface, Phys. Rev. Lett. 38 (1977), 993–996., DOI 10.1103/PhysRevLett.38.993
C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007. English edition; Translated from the French by Leila Schneps.