Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

We compute the average characteristic polynomial of Hermitised product M real or complex non-Hermitian Wigner matrices size N×N with i.i.d. matrix elements, and a such times conjugate matrix. Surprisingly, results agree that Ginibre at finite-N, which have Gaussian entries. For latter yields orthogonal for singular values matrix, whereas two polynomials involves kernel eigenvalues. This extends result Forrester Gamburd one single random only depends on first moments. In limit M→∞ fixed N we determine locations zeros polynomial, rescaled as Lyapunov exponents by taking logarithm Mth root. The position jth zero agrees asymptotically large-j exponent products matrices, hinting universality latter.