# Analytical expression for density of random matrix level ratios

Consider a hermitian matrix $$H$$ with eigenvalues $$E_{i-1}. The level spacings are defined as $$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

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 $$[0,1]$$.