Limit shapes, real and imagined

Okounkov, Andrei

doi:10.1090/bull/1512

Limit shapes, real and imagined

Author: Andrei Okounkov
Journal: Bull. Amer. Math. Soc. 53 (2016), 187-216
MSC (2010): Primary 60F10, 81T13
DOI: https://doi.org/10.1090/bull/1512
Published electronically: August 20, 2015
MathSciNet review: 3474306
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Abstract: This is an introductory discussion of limit shapes, in particular for random partitions and stepped surfaces, and of their applications to supersymmetric gauge theories.

References

M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. MR 721448, DOI https://doi.org/10.1016/0040-9383%2884%2990021-1
Thierry Bodineau, Roberto H. Schonmann, and Senya Shlosman, 3D crystal: how flat its flat facets are?, Comm. Math. Phys. 255 (2005), no. 3, 747–766. MR 2135451, DOI https://doi.org/10.1007/s00220-004-1283-4

A. Borodin, V. Gorin, and A. Guionnet, Gaussian asymptotics of discrete $\beta$-ensembles, arXiv:1505.03760.

Alexander Braverman, Instanton counting via affine Lie algebras. I. Equivariant $J$-functions of (affine) flag manifolds and Whittaker vectors, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 113–132. MR 2095901

A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential, math.AG/0409441.

R. Cerf, The Wulff crystal in Ising and percolation models, Lecture Notes in Mathematics, vol. 1878, Springer-Verlag, Berlin, 2006. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; With a foreword by Jean Picard. MR 2241754

Raphaël Cerf and Richard Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Comm. Math. Phys. 222 (2001), no. 1, 147–179. MR 1853867, DOI https://doi.org/10.1007/s002200100505

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 https://doi.org/10.1090/S0894-0347-00-00355-6

Henry Cohn, Michael Larsen, and James Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998), 137–165. MR 1641839

Eric D’Hoker and D. H. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, Theoretical physics at the end of the twentieth century (Banff, AB, 1999) CRM Ser. Math. Phys., Springer, New York, 2002, pp. 1–125. MR 1882556

R. Dobrushin, R. Kotecký, and S. Shlosman, Wulff construction, Translations of Mathematical Monographs, vol. 104, American Mathematical Society, Providence, RI, 1992. A global shape from local interaction; Translated from the Russian by the authors. MR 1181197

Roland L. Dobrushin and Senya B. Shlosman, Droplet condensation in the Ising model: moderate deviations point of view, Probability and phase transition (Cambridge, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 420, Kluwer Acad. Publ., Dordrecht, 1994, pp. 17–34. MR 1283173

S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726

Nick Dorey, Timothy J. Hollowood, Valentin V. Khoze, and Michael P. Mattis, The calculus of many instantons, Phys. Rep. 371 (2002), no. 4-5, 231–459. MR 1941102, DOI https://doi.org/10.1016/S0370-1573%2802%2900301-0

Timothy Hollowood, Amer Iqbal, and Cumrun Vafa, Matrix models, geometric engineering and elliptic genera, J. High Energy Phys. 3 (2008), 069, 81. MR 2391051, DOI https://doi.org/10.1088/1126-6708/2008/03/069

Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870

Dmitry Ioffe and Senya Shlosman, Ising model fog drip: the first two droplets, In and out of equilibrium. 2, Progr. Probab., vol. 60, Birkhäuser, Basel, 2008, pp. 365–381. MR 2477391, DOI https://doi.org/10.1007/978-3-7643-8786-0_18

D. Ioffe and S. Shlosman, Ising model fog drip: the puddle, in preparation.

P. W. Kasteleyn, Graph theory and crystal physics, Graph Theory and Theoretical Physics, Academic Press, London, 1967, pp. 43–110. MR 0253689

Richard Kenyon, Height fluctuations in the honeycomb dimer model, Comm. Math. Phys. 281 (2008), no. 3, 675–709. MR 2415464, DOI https://doi.org/10.1007/s00220-008-0511-8

Richard Kenyon and Andrei Okounkov, Limit shapes and the complex Burgers equation, Acta Math. 199 (2007), no. 2, 263–302. MR 2358053, DOI https://doi.org/10.1007/s11511-007-0021-0

Richard Kenyon, Andrei Okounkov, and Scott Sheffield, Dimers and amoebae, Ann. of Math. (2) 163 (2006), no. 3, 1019–1056. MR 2215138, DOI https://doi.org/10.4007/annals.2006.163.1019

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), no. 2, 206–222. MR 1417317, DOI https://doi.org/10.1016/0001-8708%2877%2990030-5

Grigory Mikhalkin, Amoebas of algebraic varieties and tropical geometry, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 257–300. MR 2102998, DOI https://doi.org/10.1007/0-306-48658-X_6

S. Miracle-Sole, Surface tension, step free energy and facets in the equilibrium crystal, J. Stat. Phys. 79, 183–214 (1995).

Salvador Miracle-Sole, Facet shapes in a Wulff crystal, Mathematical results in statistical mechanics (Marseilles, 1998) World Sci. Publ., River Edge, NJ, 1999, pp. 83–101. MR 1886243

Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344

Hiraku Nakajima and K\B{o}ta Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005), no. 2, 313–355. MR 2199008, DOI https://doi.org/10.1007/s00222-005-0444-1

Hiraku Nakajima and K\B{o}ta Yoshioka, Lectures on instanton counting, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 31–101. MR 2095899, DOI https://doi.org/10.1090/crmp/038/02

Hiraku Nakajima and K\B{o}ta Yoshioka, Instanton counting on blowup. II. $K$-theoretic partition function, Transform. Groups 10 (2005), no. 3-4, 489–519. MR 2183121, DOI https://doi.org/10.1007/s00031-005-0406-0

Nikita A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5, 831–864. MR 2045303

N. Nekrasov, in preparation.

Nikita A. Nekrasov and Andrei Okounkov, Seiberg-Witten theory and random partitions, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 525–596. MR 2181816, DOI https://doi.org/10.1007/0-8176-4467-9_15

N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional $N=2$ quiver gauge theories, arXiv:1211.2240

N. Nekrasov, V. Pestun, and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689.

Nikita Nekrasov and Sergey Shadchin, ABCD of instantons, Comm. Math. Phys. 252 (2004), no. 1-3, 359–391. MR 2104883, DOI https://doi.org/10.1007/s00220-004-1189-1

B. Nienhuis, H. J. Hilhorst, and H. W. J. Blöte, Triangular SOS models and cubic-crystal shapes, J. Phys. A 17 (1984), no. 18, 3559–3581. MR 772340

Andrei Okounkov, The uses of random partitions, XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2005, pp. 379–403. MR 2227852

Andrei Okounkov, Random surfaces enumerating algebraic curves, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 751–768. MR 2185779

Andrei Okounkov, Random partitions and instanton counting, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 687–711. MR 2275703

A. Okounkov, Noncommutative geometry of random surfaces, arXiv:0907.2322.

Mikael Passare and Hans Rullgård, Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope, Duke Math. J. 121 (2004), no. 3, 481–507. MR 2040284, DOI https://doi.org/10.1215/S0012-7094-04-12134-7

N. Seiberg and E. Witten, Erratum: “Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory”, Nuclear Phys. B 430 (1994), no. 2, 485–486. MR 1303306, DOI https://doi.org/10.1016/0550-3213%2894%2900449-8

N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD, Nuclear Phys. B 431 (1994), no. 3, 484–550. MR 1306869, DOI https://doi.org/10.1016/0550-3213%2894%2990214-3

M. L. Sodin and P. M. Yuditskiĭ, Functions that deviate least from zero on closed subsets of the real axis, Algebra i Analiz 4 (1992), no. 2, 1–61 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 201–249. MR 1182392

S. B. Shlosman, The Wulff construction in statistical mechanics and combinatorics, Uspekhi Mat. Nauk 56 (2001), no. 4(340), 97–128 (Russian, with Russian summary); English transl., Russian Math. Surveys 56 (2001), no. 4, 709–738. MR 1861442, DOI https://doi.org/10.1070/RM2001v056n04ABEH000417

Morikazu Toda, Theory of nonlinear lattices, Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin-New York, 1981. Translated from the Japanese by the author. MR 618652

Karen K. Uhlenbeck, The Chern classes of Sobolev connections, Comm. Math. Phys. 101 (1985), no. 4, 449–457. MR 815194

A. M. Vershik, Statistical mechanics of combinatorial partitions, and their limit configurations, Funktsional. Anal. i Prilozhen. 30 (1996), no. 2, 19–39, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 30 (1996), no. 2, 90–105. MR 1402079, DOI https://doi.org/10.1007/BF02509449

A. Vershik and S. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Soviet Math. Dokl. 18, 1977, 527–531.

A. M. Vershik and S. V. Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96 (Russian). MR 783703

Edward Witten, Dynamics of quantum field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 1119–1424. MR 1701615