Random matrix model for disordered conductors |
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Authors: | Zafar Ahmed Sudhir R Jain |
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Institution: | (1) Nuclear Physics Division, Bhabha Atomic Research Centre, 400 085 Mumbai, India;(2) Theoretical Physics Division, Bhabha Atomic Research Centre, 400 085 Mumbai, India |
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Abstract: | We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp−log2(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is
studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a
non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here).
An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed
and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever
the confining potential is given by a transcendental function for which there exist orthogonal polynomials. |
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Keywords: | Disordered conductors random matrix theory Dyson’ s Coulomb gas model |
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