Distribution of logarithmic spectra of the equilibrium energy

Let $L$ be a big invertible sheaf on a complex projective variety, equipped with two continuous metrics. We prove that the distribution of the eigenvalues of the transition matrix between the $L^2$ norms on $H^0(X,nL)$ with respect to the two metriques converges (in law) as $n$ goes to infinity to a Borel probability measure on $\mathbb R$. This result can be thought of as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson's results to the more general context of graded linear series and a more general class of line bundles. Our approach also enables us to obtain a $p$-adic analogue of our main result.