Lyapunov exponents for truncated unitary and Ginibre matrices

Abstract

In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced 'picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on GLn(C), analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.

Extremal singular values of random matrix products and Brownian motion on GL(N,C)

“…While these Lyapunov exponents are not universal, we expect that under general assumptions as N → ∞ the largest Lyapunov exponents should converge to picket fence statistics upon properly rescaling and translating. Results of this type were established for products of truncated unitary and Ginibre matrices in [AVP21]. Likewise, Theorem 1.4 also suggests that for N M , the largest singular values of Y (N ) (M ) should converge to GUE statistics given by the Airy point process.…”