A Note on the Eigenvalue Density of Random Matrices
@article{Kiessling1999ANO, title={A Note on the Eigenvalue Density of Random Matrices}, author={Michael K.-H. Kiessling and Herbert Spohn}, journal={Communications in Mathematical Physics}, year={1999}, volume={199}, pages={683-695} }
Abstract:The distribution of eigenvalues of N×N random matrices in the limit N→∞ is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.
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References
SHOWING 1-10 OF 24 REFERENCES
On the statistical mechanics approach in the random matrix theory: Integrated density of states
- Mathematics
- 1995
We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This…
Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles
- Mathematics
- 1997
This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral…
Fluctuation formula for complex random matrices
- Mathematics
- 1998
A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case where the statistic is rotationally invariant. For a general linear statistic without this…
On the Distribution of the Roots of Certain Symmetric Matrices
- Mathematics, Computer Science
- 1958
The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
Statistical Ensembles of Complex, Quaternion, and Real Matrices
- Mathematics
- 1965
Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the…
A Brownian‐Motion Model for the Eigenvalues of a Random Matrix
- Physics
- 1962
A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact…
Correlation Functions for Eigenvalues of Real Quaternian Matrices
- Mathematics
- 1966
The eigenvalue density, the two‐, and the three‐point correlation functions for the ensemble of real quaternion matrices are calculated. The forms suggest a generalization for the n‐point correlation…
Statistical mechanics of the isothermal lane-emden equation
- Mathematics
- 1982
For classical point particles in a box Λ with potential energy H(N)=N−1(1/2) ∑i≠j=1NV(xi,xj) we investigate the canonical ensemble for largeN. We prove that asN→∞ the correlation functions are…
Statistical Theory of the Energy Levels of Complex Systems. I
- Physics
- 1962
New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail,…