Limit shapes, real and imagined

Okounkov, Andrei

doi:10.1090/bull/1512

Limit shapes, real and imagined
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Bull. Amer. Math. Soc. 53 (2016), 187-216 Request permission

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 10.1016/0040-9383(84)90021-1
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 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, DOI 10.1090/crmp/038/04
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 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 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, DOI 10.1090/mmono/104
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 10.1016/S0370-1573(02)00301-0
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 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, DOI 10.1007/978-3-663-11624-0
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 10.1007/978-3-7643-8786-0_{1}8
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 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 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 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 10.1016/0001-8708(77)90030-5
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 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, DOI 10.1090/ulect/018
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 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 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 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, DOI 10.4310/ATMP.2003.v7.n5.a4
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 10.1007/0-8176-4467-9_{1}5
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 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, DOI 10.1088/0305-4470/17/18/025
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 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 10.1016/0550-3213(94)00449-8
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 10.1016/0550-3213(94)90214-3
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 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, DOI 10.1007/978-3-642-96585-2
Karen K. Uhlenbeck, The Chern classes of Sobolev connections, Comm. Math. Phys. 101 (1985), no. 4, 449–457. MR 815194, DOI 10.1007/BF01210739
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 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