Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles

TL;DR

This paper investigates how the distribution of the lowest eigenvalue in finite bosonic systems with k-body interactions transitions from Gaussian to Gumbel to Tracy-Widom distributions as k varies, revealing complex spectral behavior.

Contribution

It introduces numerical analysis of eigenvalue distributions in bosonic random matrix ensembles, identifying distribution transitions and proposing ansatz for their centroids and variances.

Findings

Distribution transitions from Gaussian to Gumbel to Tracy-Widom with increasing k.

Spacing distribution transitions from Wigner to Poisson and back to Wigner as k varies.

Eigenvalue density fits q-normal distribution, capturing transition behavior.

Abstract

We present numerical investigations demonstrating the result that the distribution of the lowest eigenvalue of finite many-boson systems (say we have $m$ number of bosons) with $k$-body interactions, modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE($k$)] and Unitary [BEGUE($k$)] random matrix Ensembles of $k$-body interactions, exhibits a smooth transition from Gaussian like (for $k = 1$) to a modified Gumbel like (for intermediate values of $k$) to the well-known Tracy-Widom distribution (for $k = m$) form. We also provide ansatz for centroids and variances of the lowest eigenvalue distributions. In addition, we show that the distribution of normalized spacing between the lowest and the next lowest eigenvalues exhibits a transition from Wigner's surmise (for $k = 1$) to Poisson (for intermediate $k$ values with $k \le m/2$) to Wigner's surmise (starting from $k = m/2$ to $k = m$) form. We analyze these transitions as a function of $q$ parameter defining $q$-normal distribution for eigenvalue densities.