Carregant...
Fitxers
Tipus de document
ArticleVersió
Versió publicadaData de publicació
Llicència de publicació
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/227113
Universality for fluctuations of counting statistics of random normal matrices
Títol de la revista
Director/Tutor
ISSN de la revista
Títol del volum
Recurs relacionat
Resum
We consider the fluctuations of the number of eigenvalues of $n \times n$ random normal matrices depending on a potential $Q$ in a given set $A$. The eigenvalues of random normal matrices are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $Q$. When $A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $N_A^{(n)}$ in $A$ has a limiting behavior given by
$$
\lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}} \operatorname{Var} N_A^{(n)}=\frac{1}{2 \pi \sqrt{\pi}} \int_{\partial_* A} \sqrt{\Delta Q(z)} d \mathcal{H}^1(z),
$$
where $\partial_* A$ is the measure theoretic boundary of $A$, $d H^1(z)$ denotes the one-dimensional Hausdorff measure, and $\Delta=\partial_z \overline{\partial_z}$. We also consider the case where $A$ is a microscopic dilation of the droplet and fully generalize a result by Akemann, Byun, and Ebke for arbitrary potentials. In this result $d \boldsymbol{H}^1(z)$ is replaced by the harmonic measure at $\infty$ associated with the exterior of the droplet. This second result is proved by strengthening results due to Hedenmalm-Wennman and Ameur-Cronvall on the asymptotic behavior of the associated correlation kernel near the droplet boundary.
Matèries (anglès)
Citació
Citació
MARZO SÁNCHEZ, Jordi, MOLAG, Leslie, ORTEGA CERDÀ, Joaquim. Universality for fluctuations of counting statistics of random normal matrices. _Journal of the London Mathematical Society-Second Series_. 2026. Vol. 113, núm. 2. [consulta: 21 de febrer de 2026]. ISSN: 0024-6107. [Disponible a: https://hdl.handle.net/2445/227113]