Limiting empirical distribution for eigenvalues of products of random rectangular matrices

We study the empirical spectral distribution of a product AN(m)=A1⋯Am of m random rectangular matrices with i.i.d. complex Gaussian entries. The product ensemble is of dimension N×N, and the rectangular matrix Aj is of size Nj×Nj+1 for j=1,…,m with Nm+1=N1=N. Explicit limit of empirical eigenvalue distribution of AN(m) is obtained in almost sure convergence as N goes to infinity. In particular, a rich feature of the limiting distributions is presented as the ratio Nj/N fluctuates for each j.