Random Riesz energies on compact Kähler manifolds
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- by Renjie Feng and Steve Zelditch PDF
- Trans. Amer. Math. Soc. 365 (2013), 5579-5604 Request permission
Abstract:
The expected Riesz energies $E_{\mu ^N_h}\mathcal E_{s}$ of the zero sets of systems of independent Gaussian random polynomials of degree $N$ are determined asymptotically as the degree $N \rightarrow \infty$ in all dimensions and codimensions. The asymptotics are proved for sections of any positive line bundle over any compact Kähler manifold.References
- Diego Armentano, Carlos Beltrán, and Michael Shub, Minimizing the discrete logarithmic energy on the sphere: the role of random polynomials, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2955–2965. MR 2775794, DOI 10.1090/S0002-9947-2011-05243-8
- Johann S. Brauchart, About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case, Integral Transforms Spec. Funct. 17 (2006), no. 5, 321–328. MR 2237493, DOI 10.1080/10652460500431859
- Brandon Ballinger, Grigoriy Blekherman, Henry Cohn, Noah Giansiracusa, Elizabeth Kelly, and Achill Schürmann, Experimental study of energy-minimizing point configurations on spheres, Experiment. Math. 18 (2009), no. 3, 257–283. MR 2555698
- B. Bergersen, D. Boal and P. Palffy-Muhoray, Equilibrium configurations of particles on a sphere: The case of logarithmic interaction, J. Phys. A: Math. Gen. 27 (1994) 2579–2586.
- Robert Berman, Sébastien Boucksom, and David Witt Nyström, Fekete points and convergence towards equilibrium measures on complex manifolds, Acta Math. 207 (2011), no. 1, 1–27. MR 2863909, DOI 10.1007/s11511-011-0067-x
- Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Poincaré-Lelong approach to universality and scaling of correlations between zeros, Comm. Math. Phys. 208 (2000), no. 3, 771–785. MR 1736335, DOI 10.1007/s002200050010
- Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395. MR 1794066, DOI 10.1007/s002220000092
- Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Correlations between zeros and supersymmetry, Comm. Math. Phys. 224 (2001), no. 1, 255–269. Dedicated to Joel L. Lebowitz. MR 1868999, DOI 10.1007/s002200100512
- Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99–148. MR 2257398, DOI 10.1090/S0894-0347-06-00546-7
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state, J. Phys. A 29 (1996), no. 5, L101–L105. MR 1383056, DOI 10.1088/0305-4470/29/5/004
- Paul Hriljac, Splitting fields of principal homogeneous spaces, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 214–229. MR 894513, DOI 10.1007/BFb0072982
- D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194. MR 2104914
- Marc Hindry and Joseph Silverman, Sur le nombre de points de torsion rationnels sur une courbe elliptique, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 2, 97–100 (French, with English and French summaries). MR 1710502, DOI 10.1016/S0764-4442(99)80469-8
- A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), no. 2, 523–538. MR 1458327, DOI 10.1090/S0002-9947-98-02119-9
- Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
- Zhiqin Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235–273. MR 1749048
- K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. MR 166188
- Theodor William Melnyk, Osvald Knop, and William Robert Smith, Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited, Canad. J. Chem. 55 (1977), no. 10, 1745–1761 (English, with French summary). MR 0444497
- E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett. 1 (1994), no. 6, 647–662. MR 1306011, DOI 10.4310/MRL.1994.v1.n6.a3
- Bernard Shiffman and Steve Zelditch, Number variance of random zeros on complex manifolds, Geom. Funct. Anal. 18 (2008), no. 4, 1422–1475. MR 2465693, DOI 10.1007/s00039-008-0686-3
- Bernard Shiffman and Steve Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200 (1999), no. 3, 661–683. MR 1675133, DOI 10.1007/s002200050544
- Gerold Wagner, On means of distances on the surface of a sphere (lower bounds), Pacific J. Math. 144 (1990), no. 2, 389–398. MR 1061328
- Gerold Wagner, On means of distances on the surface of a sphere. II. Upper bounds, Pacific J. Math. 154 (1992), no. 2, 381–396. MR 1159518
- Steve Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317–331. MR 1616718, DOI 10.1155/S107379289800021X
- Steve Zelditch and Qi Zhong, Addendum to “Energies of zeros of random sections on Riemann surfaces”. Indiana Univ. Math. J. 57 (2008), No. 4, 1753–1780 [MR2440880], Indiana Univ. Math. J. 59 (2010), no. 6, 2001–2005. MR 2919745, DOI 10.1512/iumj.2010.59.59073
- Qi Zhong, Energies of zeros of random sections on Riemann surfaces, Indiana Univ. Math. J. 57 (2008), no. 4, 1753–1780. MR 2440880, DOI 10.1512/iumj.2008.57.3329
Additional Information
- Renjie Feng
- Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0G4
- MR Author ID: 939975
- Email: renjie@math.mcgill.ca
- Steve Zelditch
- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 186875
- Email: zelditch@math.northwestern.edu
- Received by editor(s): January 11, 2012
- Received by editor(s) in revised form: April 23, 2012
- Published electronically: March 12, 2013
- Additional Notes: This research was partially supported by NSF grant #DMS-0904252.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5579-5604
- MSC (2010): Primary 58J37, 32L81, 32A60, 60D05
- DOI: https://doi.org/10.1090/S0002-9947-2013-05870-9
- MathSciNet review: 3074383