# Small gaps of GOE

## TL;DR

This paper investigates the distribution of the smallest gaps in the Gaussian orthogonal ensemble, showing they tend to a Poisson distribution after normalization, with explicit density formulas for the k-th smallest gaps.

## Contribution

It provides a rigorous analysis of the limiting distribution of the smallest gaps in GOE, including explicit density formulas for the normalized gaps.

## Key findings

- Smallest gaps normalized by n tend to a Poisson distribution.
- Explicit density of the k-th normalized smallest gap is derived.
- Results deepen understanding of eigenvalue spacing in GOE.

## Abstract

In this article, we study the smallest gaps of the Gaussian orthogonal ensemble. The main result is that the smallest gaps, after normalized by $n$, will tend to a Poisson distribution, and the limiting density of the $k$-th normalized smallest gaps is $ 2{}x^{2k-1}e^{-x^{2}}/(k-1)!$.