  An exchangeable random matrix is a random matrix with distribution invariant
under any permutation of the entries. For such random matrices, we show, as the
dimension tends to infinity, that the empirical spectral distribution tends to
the uniform law on the unit disc. This is an instance of the universality
phenomenon known as the circular law, for a model of random matrices with
dependent entries, rows, and columns. It is also a non-Hermitian counterpart of
a result of Chatterjee on the semi-circular law for random Hermitian matrices
with exchangeable entries. The proof relies in particular on a reduction to a
simpler model given by a random shuffle of a rigid deterministic matrix, on
Hermitization, and also on combinatorial concentration of measure and
combinatorial Central Limit Theorem. A crucial step is a polynomial bound on
the smallest singular value of exchangeable random matrices, which may be of
independent interest.
