Perspective

Statistics of complex eigenvalues of large random matrices have recently attracted much interest in both physical and mathematical communities.

The mathematical treatment of random matrices with no symmetry conditions imposed goes back to the pioneering work by Ginibre [1]. Until recently, the progress in the field has been slow but steady [2,3,4,5].

For a long period, physicists have used nonhermitean random matrices to describe the generic statistical properties of resonances in open quantum chaotic systems (for a detailed bibliography see [6]) and to characterize typical features of dissipative chaotic quantum maps [7]. Complex and real asymmetric random matrices also enter in the study of the chaotic dynamics of asymmetric neural networks [8].

A boost of interest in nonhermitean random matrices occurred recently in several fields. In quantum chromodynamics it was realized [9] that they could be a useful tool for the understanding of generic features of spontaneous chiral symmetry breaking at nonzero baryon density. Unexpected results were obtained in the study of non-symmetric random matrix models for the depinning of vortices in disordered superconductors [10]. Complex spectra of random matrices also emerge in studies of the advection of a passive scalar by a quenched random velocity field [11], and can be used to extract information on correlations in two-dimensional classical plasmas [12]. In addition, they might provide an insight into two-dimensional Calogero-Sutherland models [13] and may have close connections with the issue of exact integrability [14]. In parallel, new types of random non-Hermitian matrices were discovered [15], and new methods were developed [6,15,16,17].

In general, one expects that non-Hermitian random matrices are of potential relevance whenever a non-selfadjoint linear operator appears in a physical problem. To this end, one should mention the problem of hydrodynamic stability and the issue of "pseudospectra" [18], the method of "complex rotation" (see e.g. [19]) , as well as various issues related to non-equilibrium statistical mechanics [20].

All these actual and potential developments make it necessary to bring together researches working in diverse fields on various aspects of non-hermitian random matrices and non-selfadjoint random operators, as well as on closely related problems such as complex roots of random polynomials [22].

Such a conference is expected to provide a fruitful exchange of ideas between different branches of physics and mathematics as well as a reasonable coordination of efforts necessary for further progress in this rapidly developing area of research.

Scope of the Conference

The objective of this conference is to bring together both physicists and mathematicians in order to provide a broad discussion on applications of non-Hermitean random matrices and non-selfadjoint random operators in various branches of theoretical physics and to review recent progress in the field.

The topics of the conference will cover: general theory of non-Hermitian random matrices, complex roots of random polynomials, resonances and time delays in quantum chaotic scattering (including the method of "complex scaling"), localisation transition in non-hermitian quantum mechanics, QCD at finite chemical potential, chaos and dissipation, hydrodynamic stability and pseudospectra, diffusion in a random velocity field, non-hermiticity and integrable models and some other topics.

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[3] P.J. Forrester, Phys. Lett. A 169,21
(1992); J. Phys. A:Math. Gen. 26, 1179 (1993).

[4] A. Edelman, J. Multivariate Anal. 60, 203
(1997), and references therein.

[5] G. Oas, Phys. Rev. E 55, 205 (1997).

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38, 1918 (1997).

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(1996); M.A. Halasz, J. Verbaarschot et al., e-preprints hep-th/9703006;
hep-th/9704007.

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(1996); K.B. Efetov preprint cond-mat/9702091; R.A.Janik et al.,
e-preprint cond-mat/9705098;
B.A. Khoruzhenko and I. Goldscheid, unpublished.

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and J. Wang, Phys. Rev. Lett. 76, 1461
(1996); J. Chalker and J. Wang, e-preprint cond-mat/9704198.

[12]P. Forrester and B. Jancovici, Int. J. Mod. Phys. A
11, 941 (1996)

[13] M. Feigelman and M. Skvortsov, e-preprint cond-mat/9703215;
A. Khare and K. Ray, e-preprint hep-th/9609025.

[14] T. Akuzawa and M. Wadati, J. Phys. Soc. Jpn. 65 ,
1583 (1996).

[15] Y.V. Fyodorov, B. Khoruzhenko and H.-J. Sommers,
Physics Letters A 226, 46 (1997); e-preprint cond-mat/9703152.

[16] R.A. Janik et al., e-preprint cond-mat/9612240; J. Feinberg and
A. Zee, e-preprint cond-mat/9702091

[17] M. Kus, F. Haake, D. Zaitsev and A.Huckleberry,
to be published.
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[18] J.Burd\"{o}rfer et al., Chaos, Solitons and Fractals
5 p.1235; R. Bl\"{u}mel, Phys. Rev. E 54, 5420 (1996).

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[20] E. Bogomolny et al., J. Stat. Phys. 85, 639 (1996).