UC Berkeley

Berezin--Toeplitz operators are quantizations of functions on K\"ahler manifolds equipped with a positive line bundle $L$. When the K\"ahler manifold $M$ is compact, the quantization procedure associates every smooth function $f\in C^\infty(M)$ to a family of matrices $T_N f $ indexed by $N\in \N$, whose size goes to infinity as $N\to \infty$ (corresponding to the semiclassical limit $h \to 0$). Each matrix $T_N f$ acts on the finite-dimensional space of holomorphic sections of the $N$th tensor power of $L$.

In this thesis we study the spectrum of $T_N f + \delta \mathcal{G}_\omega(N)$ where $\delta = \delta (N) > 0$ and $\mathcal{G}_\omega(N)$ is a family of random matrices. Under certain conditions, the spectrum satisfies a probabilistic Weyl law involving $f$ and the volume form on the K\"ahler manifold. Specifically, as $N\to \infty$, the (normalized) empirical spectral distribution of $T_N f + \delta \mathcal{G}_\omega(N)$ converges weakly almost surely to the (normalized) push-forward by $f$ of the Liouville volume form on $M$. This generalizes a result of Martin Vogel, which considered the case of torii.

Proving this result requires extending the usual calculus of Berezin--Toeplitz operators to an exotic class of functions.The exotic nature of these functions (classical observables) refers to the property that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sj\"ostrand’s method of complex stationary phase. For this more exotic class of functions, we obtain a functional calculus result, a trace formula, and a parametrix construction.