Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying distribution of $r_i$ bounded let us consider $\tilde{r}_i = \min(r_i, 1/r_i)$.

Now, let $H$ be random and distributed according to the Gaussian orthogonal ensemble. Then the density of the eigenvalues of H, $E_i$, is well known. An approximation for the distribution of $s_i$ is known as Wigner surmise. The approximation is exact for 2x2 matrices and still is a good approximation in the large matrix size limit. In quantum chaos, the level spacings of most Hamiltonians $H$, which are considered chaotic, follow the distribution of a random Gaussian matrix.

In https://arxiv.org/abs/1212.5611 an approximation for $r_i$ is derived. The idea is the same idea as Wigner's: The formula is exact for 3x3 matrices and still a good approximation in the large matrix limit.

Is there an approximation known for the distribution of $\tilde{r}_i = \min(r_i, 1/r_i)$ as well? The above paper mentions an exact result for the expected value of $\tilde{r}_i$ but not a closed form expression for its density.


1 Answer 1


I should have read the paper I cited more carefully as it states the answer to my question. According to https://arxiv.org/abs/1212.5611:

Let $P(r)$ denote the density of $r_i$ defined above. Then it holds that $$ P(r) = \frac{1}{r^2}P(1/r), $$ according to some symmetry arguments described in https://arxiv.org/abs/1212.5611. Therefore the density of $\tilde{r}_i$ is given by $$ \tilde{P}(r) = 2P(r), $$ where the support of $\tilde{P}$ is the interval $[0,1]$.


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