Some clarification of infinitesimal derivation in random matrix theory

Today I looked at Ben Arous and Bourgade’s paper on extreme eigenvalue gaps and in the proof of the largest gap they referred to Mehta’s classic treatise on random matrix theory Appendix 8. In there, he clarified the relation between three density quantities, $E_n(s)$, $\tilde{F}_n(s)$ and $p_n(s)$. Here are the meaning of these quantities: $E_n(s)$ is simply the probability that a randomly chosen interval of length $s$ contains no eigenvalues (or what he calls spacings for more general point processes on $\mathbb{R}$). $\tilde{F}_n(s)$ is the probability density that a random interval of size $s$ has no eigenvalues, but exactly one eigenvalue lies at exactly the right boundary of this interval. Finally $p_n(s)$ is the density that there are no eigenvalues in a random interval of length $s$, but there are exactly two eigenvalues lying at the left and right endpoints of this interval respectively.

Here are the paraphrase of Mehta’s original explanation, which was difficult to isolate from the rest of the book: $E_n(s)$ was interpreted the same way as above, but $\tilde{F}_n(s)ds$ is the probability that a interval of size s with left endpoint at 0 has no eigenvalues, but the same interval with an additional length of $ds$ to the right has one eigenvalue. For $p_n(s)$, a definition was not even given at first, but he later explained that $p_n(s)ds$ is the probability of the spacing between consecutive eigenvalues chosen around 0 being in the interval $[s,s+ds]$.

So given the two interpretations above, we can easily infer their relationships based on calculus of infinitesimals:

1. $\frac{d}{ds} E_n(s) = - \tilde{F}_n(s)$, which is because $E_n(s) - E_n(s+ds)$ is the probability that no eigenvalue lies in the interval of length s, but one eigenvalue lies in the additional $ds$ interval to the right.

2. $\frac{d}{ds} \tilde{F}_n(s) = -p_n(s)$, which is because $\tilde{F}_n(s) - \tilde{F}_n(s+ds)$ is the probability that an eigenvalue lies at the right endpoint of the interval of length s, the interval of length s has no eigenvalues, and the negation of the event that the interval $[-ds,s]$ has no eigenvalues, and an eigenvalue lies at the left endpoint of that interval. This clearly gives the probability of empty interval $[0,s]$ plus two eigenvalues at the two endpoints. The reason we choose to extend to the left here is that infinitesimal quantities cannot be the base for another infinitesimal increment. But this is not a perfectly convincing reason, hence the appendix did not go over the detail of the second relation either. It remains to find a self-consistent explanation for why extending to the left and to the right give the same relation 2.