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In Random Matrix Theory,

Question 1):

Two Distributions are very popular.

1) The density of Energy levels $\rho(E)$.

2) Distribution of nearest energy levels spacing $p(s)$.

I understand why first is so useful/popular, From $\rho(E)$, one can calculate the entropy of the system by $S=-\ln(\rho(E))$.

I am finding difficult to understand why $p(s)$ is so popular ?. what can be inferred from a $p(s)$ other than linear/quadratic repulsion ?. How it is useful ?.

I understand, this is a general question, Could anyone through some light on it.

Question 2):

$p(s)$ for Gaussian Orthogonal ensemble is given as, $p(s)$ =$\frac{\pi}{2} s \exp^{-\frac{\pi}{2} s^2}$. From this It is clear that $p(s)\rightarrow 0$, as $s \rightarrow 0$ $\Rightarrow$ NO DEGENERACY!!

This $p(s)$ is valid for a large number of the systems, and none of them can have degeneracy ??, Can anybody help me in understanding this ??

It would be a much help to me.

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1 Answer 1

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Question 1:

There are many uses to the spacing of eigenvalues. If you model a complex Hamiltonian as a random matrix, the absorption and emission spectra are related to the spacing between eigenvalues, not to their density. Indeed, Wigner's original motivation in the study of random matrices was to understand the spectra of Heavy nuclei.

More generally, random matrix spacing should be universal for quantum chaotic systems, whereas the eigenvalue spacing in integrable systems exhibit Poisson statistics.

Another important use is to understand the spectral gap: the spacing between the 1st and 2nd eigenvalues (this has a different distribution than the spacing in the bulk).

Random matrix eigenvalue spacings also appear in problems not directly related to random matrices. For example, some interacting particle systems follow random matrix statistics, and there $p(s)$ describes the real distance between particles. A famous example is the statistics of busses in Cuernavaca Mexico. Another non-matrix example is the zeros of the zeta function, whose spacings seem to follow random matrix statistics.

Question 2:

Indeed, in many ensembles of random matrices, the probability of getting a degenerate eigenvalue by chance is zero. Note that matrices that pop up in applications are not random, they have structure which could lead to degeneracy. Usually, degeneracy is a result of some symmetry property of the given matrix, and often cases of "accidental degeneracy" can be shown to actually result from some "hidden" symmetry.

Note that your equation for $p(s)$, known as Wigner's surmise, describes the eigenvalue spacing of a random 2 x 2 matix. For the spacing of random N x N matrices the formula for the spacing distribution is more complicated (but the function is qualitatively similar to Wigner's surmise).

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