The lattice fermion determinants, in a given background gauge field, are evaluated for two different kinds of random lattices and compared to those of naive and Wilson fermions in the continuum limit. While the fermion doubling is confirmed on one kind of lattices, there is positive evidence that it may be absent for the other, at least for vector interactions in two dimensions. Combined with previous studies, arbitrary randomness by itself is shown to be not a sufficient condition to remove the fermion doublers. ((orig.)).