  We study local correlations of certain interacting particle systems on the
real line which show repulsion similar to eigenvalues of random Hermitian
matrices. Although the new particle system does not seem to have a natural
spectral or determinantal representation, the local correlations in the bulk
coincide in the limit of infinitely many particles with those known from random
Hermitian matrices; in particular they can be expressed as determinants of the
so-called sine kernel. These results may provide an explanation for the
appearance of sine kernel correlation statistics in a number of situations
which do not have an obvious interpretation in terms of random matrices.
