Constrained random matrices

Johan Stäring, B. Mehlig

Chalmers University of Technology and Göteborg University, Sweden

The study of random matrices [1] has provided insight into many physical problems both in the quantum as well as in the classical domain. For example, random matrices have been very successfully used to model statistical properties of disordered conductors and of highly excited classically chaotic quantum systems. In the classical domain, random matrices arise in the context of diffusion in random, directed environments (see [2] and references therein). In all of these cases the random matrix elements obey symmetry requirements where appropriate, and are otherwise taken to be independently distributed random variables.

There are many cases, however, where constrained matrices must be considered: examples are electron hopping in amorphous semiconductors (see [3] and references therein), random reactance networks [4], and random Master equations.

We calculate the density of eigenvalues of constrained random $N \times N$ matrices in the limit of large $N$ using a perturbative expansion [5]. The results are compared to the results of numerical simulations for finite values of $N$.

References
[1] M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels (Academic Press, New York, 1991).
[2] J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. 79, 1797 (1997).
[3] M. Mezard, G. Parisi, and A. Zee, Nucl. Phys. B 559, 689 (1999).
[4] Y. Fyodorov, J. Phys. A-Math. Gen. 32, 7429, (1999).
[5] B. Mehlig and J. Chalker, J. Math. Phys. 41, 3233 (2000).