Szegő asymptotics for multiple orthogonal polynomials with respect to Angelesco weights

The system of Angelesco weights: $\{\rho_j(x), \, x\in \Delta_j \subset \mathbb{R}\}_{j=1}^d$, where segments $\Delta_j$: $\Delta_j\cap \Delta_k=\varnothing$, $j\neq k$, is one of the basic systems for multiple orthogonal polynomials (MOPs) $Q_{\vec{n}}$ indexed by $\vec{n}:=\{n_j\}_{j=1}^d$:
$$ \int Q_{\vec{n}}(x)\,x^k\, \rho_j(x)\,dx=0 ,\qquad k=1,...,n_j,\quad j=1,...,d,$$
where $\deg Q_{\vec{n}}= |\vec{n}|:=n_1+...+n_d$ . It is clear that for $d=1$ this situation reduces to $Q_{{n}}$ – orthogonal polynomials (OPs).

In the talk, we start with Widom's approach to strong (or Szegő type) asymptotics for OPs, then discuss an adaptation of this approach for MOPs with respect to Angelesco system: a known partial result when $d=2, \vec{n} = (n,n)$, and perspectives for the general case motivated by modern requests from spectral problems for Schrödinger operators on the Cayley tree graph.