A natural generalization of the Lebesgue sequence space to the non-commutative setting of n by n-matrices is to take the p-norm of the singular values, the Schatten p-norm. Thus, in case of p equal to one we obtain the nuclear/trace norm, in case of p equal to two the Hilbert-Schmidt/Frobenius norm and in case of p equal to infinity the spectral norm. In this talk, we will be interested in the volume of Schatten p-balls, that is, unit balls with respect to Schatten p-norms, as their dimension tends to infinity. In contrast to the commutative analog, the finite-dimensional Lebesgue sequence space for which the volume of the unit ball is known since Dirichlet, exact results are only available for p equal to two and to infinity. Asymptotics for general p have been established starting in the 80's with two very recent contributions in the last month. We give insight on underlying connections to the theory of random matrices, logarithmic potentials and statistical mechanics. In particular, we highlight a large deviation principle due to Leblé and Serfaty from which we obtain finer asymptotics for the volume of Schatten $p$-balls in the self-adjoint case. Perhaps interestingly, both the differential entropy and its noncommutative analog - the free entropy - of the corresponding equilibrium measure appear.