Truncated linear statistics associated with the top eigenvalues of random
matrices
Given an invariant random matrix ensemble with a joint distribution of
eigenvalues $P(\lambda_1,\lambda_2,\ldots,\lambda_N)$, we study the
distribution of a truncated linear statistics (TLS):
$\tilde{L}=\sum_{n=1}^K f(\lambda_n)$ associated with the top $K$ ordered
eigenvalues $\lambda_1>\lambda_2\ldots>\lambda_N$. Motivated by the
statistical physics of fluctuating non-intersecting Brownian interfaces on a
solid substrate, we focus on the Laguerre ensemble with
$f(\lambda)=\sqrt{\lambda}$. Using a Coulomb gas method, we derive
the distribution of $\tilde{L}$ in the limit when $K\to \infty$, $N\to
\infty$ but with the fraction $0< \kappa= K/N<1$ fixed. We show that the
constraint that $\tilde{L}= \sum_{n=1}^K f(\lambda_n)$ is fixed,
leads to an infinite order phase transition in the underlying Coulomb
gas. This transition corresponds to a change in the optimal charge density
in the Coulomb gas, from a density supported on two disjoint intervals (for
large $\tilde{L}$) to a density on a single interval (for small
$\tilde{L}$). In the small $\tilde{L}$ phase, the density has a
logaritmic divergence inside the bulk. Assuming that $f(\lambda)$ is
monotonous, we show that these features arise for any random matrix
ensemble and any TLS $f(\lambda)$, which makes our results robust and
universal.