Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature

Ricardo Marino, Satya N. Majumdar, Gregory Schehr +1

Let ${\mathcal{P}}_{\beta }^{(V)}(N_{\mathcal{I}})$ be the probability that a $N\times N$ $\beta$-ensemble of random matrices with confining potential $V(x)$ has $N_{\mathcal{I}}$ eigenvalues inside an…