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Random normal matrices and Ward identities. (English) Zbl 1388.60020
Summary: We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman’s solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

MSC:
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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[1] Ameur, Y. (2013). Near-boundary asymptotics for correlation kernels. J. Geom. Anal. 23 73-95. · Zbl 1267.30027
[2] Ameur, Y., Hedenmalm, H. and Makarov, N. (2010). Berezin transform in polynomial Bergman spaces. Comm. Pure Appl. Math. 63 1533-1584. · Zbl 1220.30005
[3] Ameur, Y., Hedenmalm, H. and Makarov, N. (2011). Fluctuations of eigenvalues of random normal matrices. Duke Math. J. 159 31-81. · Zbl 1225.15030
[4] Ameur, Y., Kang, N.-G. and Makarov, N. (2014). Rescaling Ward identities in the random normal matrix model. Available at . arXiv:1410.4132
[5] Berman, R. (2008). Determinantal point processes and fermions on complex manifolds: Bulk universality. Preprint. Available at . arXiv:0811.3341
[6] Berman, R., Berndtsson, B. and Sjöstrand, J. (2008). A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Mat. 46 197-217. · Zbl 1161.32001
[7] Berman, R. J. (2009). Bergman kernels for weighted polynomials and weighted equilibrium measures of \(\mathbb{C}^{n}\). Indiana Univ. Math. J. 58 1921-1946. · Zbl 1175.32002
[8] Borodin, A. (2011). Determinantal point processes. In The Oxford Handbook of Random Matrix Theory 231-249. Oxford Univ. Press, Oxford. · Zbl 1238.60055
[9] Borodin, A. and Sinclair, C. D. (2009). The Ginibre ensemble of real random matrices and its scaling limits. Comm. Math. Phys. 291 177-224. · Zbl 1184.82004
[10] Elbau, P. and Felder, G. (2005). Density of eigenvalues of random normal matrices. Comm. Math. Phys. 259 433-450. · Zbl 1129.82017
[11] Hedenmalm, H. and Makarov, N. (2013). Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. (3) 106 859-907. · Zbl 1336.82010
[12] Hedenmalm, H. and Shimorin, S. (2002). Hele-Shaw flow on hyperbolic surfaces. J. Math. Pures Appl. (9) 81 187-222. · Zbl 1031.35152
[13] Hörmander, L. (1994). Notions of Convexity. Progress in Mathematics 127 . Birkhäuser, Boston, MA. · Zbl 0835.32001
[14] Johansson, K. (1988). On Szegő’s asymptotic formula for Toeplitz determinants and generalizations. Bull. Sci. Math. (2) 112 257-304. · Zbl 0661.30001
[15] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151-204. · Zbl 1039.82504
[16] Mehta, M. L. (2004). Random Matrices , 3rd ed. Pure and Applied Mathematics ( Amsterdam ) 142 . Elsevier, Amsterdam. · Zbl 1107.15019
[17] Rider, B. and Virág, B. (2007). The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN 2007 Art. ID rnm006. · Zbl 1130.60030
[18] Saff, E. B. and Totik, V. (1997). Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 316 . Springer, Berlin. · Zbl 0881.31001
[19] Sakai, M. (1991). Regularity of a boundary having a Schwarz function. Acta Math. 166 263-297. · Zbl 0728.30007
[20] Simon, B. (2008). The Christoffel-Darboux kernel. In Perspectives in Partial Differential Equations , Harmonic Analysis and Applications. Proc. Sympos. Pure Math. 79 295-335. Amer. Math. Soc., Providence, RI. · Zbl 1159.42020
[21] Soshnikov, A. (2000). Determinantal random point fields. Uspekhi Mat. Nauk 55 107-160. · Zbl 0991.60038
[22] Wiegmann, P. and Zabrodin, A. (2003). Large scale correlations in normal non-Hermitian matrix ensembles. J. Phys. A 36 3411-3424. · Zbl 1039.65034
[23] Wiegmann, P. and Zabrodin, A. (2006). Large \(N\) expansion for normal and complex matrix ensembles. In Frontiers in Number Theory , Physics , and Geometry I 213-229. Springer, Berlin. · Zbl 1163.82005
[24] Zabrodin, A. (2006). Matrix models and growth processes: From viscous flows to the quantum Hall effect. In Applications of Random Matrices in Physics. NATO Sci. Ser. II Math. Phys. Chem. 221 261-318. Springer, Dordrecht. · Zbl 1136.82367
[25] Zabrodin, A. and Wiegmann, P. (2006). Large-\(N\) expansion for the 2D Dyson gas. J. Phys. A 39 8933-8963. · Zbl 1098.82011
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