5 problems

An ensemble of random matrices $A^N$ has independent centered $L^2$ entries, each with variance $1/N$. Girko's circular law conjecture. The empirical measure of the eigenvalues of…

Let $\mathcal{M}$ be the finite von Neumann algebra considered in the paper, with its tracial $L^2$-norm, and let $T\in\mathcal{M}$. Write $\widetilde{T}^{(n)}$ for the $n$th Aluth…

Let $(\mathscr{A},\tau)$ be a $W^*$-probability space and let $b_0$ be an invertible operator in it. For $z\in\mathbb{C}$, define … For any operator $a$, write $|a|^2=a^*a$. Define…

For $t>0$, let $B_t^N$ be Brownian motion on $\mathsf{GL}(N;\mathbb{C})$, and let $b_t$ be the free multiplicative Brownian motion with Brown measure $\mu_{b_t}$. The empirical eig…

Brown measure convergence conjecture. The eigenvalue distributions $\mu_{A_N}$ converge to the Brown measure $\mu_x$ of $x$.