The Widom conjecture concerns the asymptotic spectral density of Toeplitz operators of the form $\Pi_U F \Pi_V F^* \Pi_U$, where $\Pi_U$ is the operator of multiplication by the indicator of an open set $U$ and $F$ is the Fourier transform, in a semclassical limit where the size of $U$ and/or $V$ tends to infinity. This conjecture was proved by Widom himself in the 80's and by A. Sobolev and his collaborators a decade ago.

Widom's initial motivation was to prove an analogue of a theorem by Basor on large Toeplitz matrices with indicator symbols, and in both cases one can translate the spectral asymptotics into probabilistic quantities for natural point process models -- for instance, Basor's result describes the number of eigenvalues of a random large unitary matrix which lie inside an interval of the unit circle.

In turns, this interpretation prompts potential generalisations of the Widom conjecture to operators built with other kinds of projectors, such as general spectral projectors for quantum hamiltonians. In this talk, I will present an overview of the Widom conjecture, the probabilistic interpretation, and my joint work with Gaultier Lambert (some of it in progress) towards the generalised Widom conjecture.