Speaker
Description
We solve exactly, for arbitrary finite channel number $N$, the Riccati dynamics of a quasi-one-dimensional random matrix Schr""odinger operator with Hermitian white-noise potential in the unitary symmetry class.
At $\beta=2$, the associated Dyson--Riccati gas admits an exact free-fermion representation, yielding explicit finite-$N$ expressions for the density of states and the total Lyapunov growth rate. Equivalently, these quantities are encoded by a Kontsevich matrix Airy $\tau$-function. We then discuss the large-$N$ spectral edge, where the finite-channel Lifshitz-type tail rounds the sharp $N=\infty$ edge within an $N^{-4/5}$ scaling window governed by a universal cubic-oscillator problem. Finally, we discuss a conjectured connection between this scaling regime and the localized spectral edge of a broad class of one-dimensional random band matrices.
The talk is based on joint work in progress with Pierre Le Doussal.