Hi, I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the unifor …

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is c …

Suppose $\mathfrak{A}$ is an algebra (over $\mathbb{C}$, let's say), $\phi$ a linear functional on $\mathfrak{A}$, and $A_1, A_2 \subset \mathfrak{A}$ subalgebras which are $\phi$- …

We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_ …

Consider the $N\times N$ matrix $$ M = \left(\begin{array} \\ 0 & 1 & & 0 \\ 1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ 0 & & …

In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperat …

Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a fre …

I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how …