67 problems

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and satisfy the assumptions denoted by $(\mathrm{gibbs})$. For $\lambda\in\mathbb{R}$, let the free Gibbs measure associated w…

For every finitely generated, torsion-free, non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, the group von Neumann algebra $L(\Gamma)$ is isomorphic to a free…

For $r\in(1,\infty]$, let $L(\mathbb F_r)$ be the interpolated free group factor, and let the first-order fundamental group of $L(\mathbb F_r)$ be the group of positive scalars $t$…

Let $p_d$ be a monic polynomial of degree $d$ with non-negative real roots … Let $\mu\in\mathcal{P}_{+,c}$, and suppose that the empirical root distributions of $p_d$ converge weak…

Let $V$ be the double-well potential … where $g$ is a negative constant sufficiently close to zero, and let $\mu_t$ and $\mu_V$ denote the empirical measure process and the equilib…

Let $V$ be a $C^2$ potential function, and let $K_1,K_2,r>0$ satisfy … and … Suppose that $\Sigma_V$ has a unique minimizer $\mu_V$ whose support is a single compact set. Convergen…

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by … For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's…

Let $f$ be in the Laguerre--Pólya class and satisfy the regular-variation condition on its roots. Let … and set $\rho=\frac{1}{\alpha}-1$. Write $\mu_p$ for the empirical root dist…

Unimodality conjecture. If $bc$ and $bd$ are symmetric unimodal distributions on $a4$, then $bc\boxtimesbd$ is also unimodal.

Let $\lambda^1(N),\lambda^2(N)\in\widehat U(N)$, for $N=1,2,\dots$, be two sequences of signatures satisfying the technical assumption of Definition $\mathrm{regularity}$, and supp…

Let $r,s$ be positive integers, let $\Phi(x,j)$ be the generating-function expression used in the preceding propositions, and write $[x^{r+1}]$ and $[y^{s+1}]$ for coefficient extr…

Let $X$ and $Y$ be colored graphs. Write $X*Y$ for the colored graph obtained by putting a copy of $X$ at each vertex of $Y$, let $\mu(Z)$ denote the spectral measure of the charac…

Let $T$ be Voiculescu's circular operator, let $\tau$ denote the trace, and let $k,n\in{\mathbf N}$. Moment formula conjecture. … The formula is proved for $k=1$, and it is also el…

Asymptotical freeness conjecture. The sequences

Diagonal monitored-transport limit. If $L_N/N\to\tau$, then

Let $\vb{A}$ be a large deterministic matrix and $\vb{B}$ a rotationally invariant random matrix. Define the matrix-valued transforms $\mathcal{G}_{\vb{A}+\vb{B}}$,…

Let $\mu$ and $\nu$ be compactly supported probability measures, let $c\in[0,\infty]$, and let $\mu\oplus_c\nu$ denote their $c$-convolution. Let $f\in C^{4}(\mathbb{R})$ satisfy…

Let $p$ and $q$ be real-rooted polynomials of degree $d$, and let $\f$ be in $C^{4}(\mathbb{R})$ with $f^{(4)}\geq 0$. Write $\mu_p$ and $\mu_q$ for the probability measures assign…

Let $\mathbf{B}$ be a large rotationally invariant random matrix, with $\mathfrak{g}_{1,\mathbf{B}}$ and $\mathfrak{g}_{2,\mathbf{B}}$ denoting the corresponding transform componen…

Let $\mathbf{A}$ and $\mathbf{B}$ be two large independent random matrices, and let $\mathbf{U}$ be a Haar-distributed unitary random matrix. For large $z\in\mathbb{C}$ or large…

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…

Airy edge universality conjecture. Under some mild conditions on the summands, the upper edge limit of $Vec{\lambda}(N)$ near $mu_+$ is still given by $\operatorname{Airy}(\beta)$…

Let $\mu$ be a probability measure and let $[\mu]_\tau$ denote its free compression for $\tau\in[0,1]$. Define the sub-probability measure $\mu_\tau$ of total mass $\tau$ by…

Let $X=(X_1,\ldots,X_n)$ be a noncommutative random vector with joint law $\mu_X$, let $V_0$ and $V_1$ be potentials, and let $A=A^{\dagger}$ be a free Stein kernel for $X$ with re…

Let $u_{\phi}$ and $u_{\psi}$ be free Gibbs measures associated with strictly $\mathcal{C}^2$ convex potentials $\phi,\psi:\mathbb{R}\rightarrow\mathbb{R}$, and write…