English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We prove that the largest and smallest root in modulus of random Kacpolynomials have a non-universal behavior. They do not converge towards theedge of the support of the limiting distribution of the zeros. Thisnon-universality is surprising as the large deviation principle for theempirical measure is universal. This is in sharp contrast with random matrixtheory where the large deviation principle is non-universal but thefluctuations of the largest eigenvalue are universal. We show that the modulusof the largest zero is heavy tailed, with a number of finite moments boundedfrom above by the behavior at the origin of the distribution of thecoefficients. We also prove that the random process of the roots of modulussmaller than one converges towards a limit point process. Finally, in the caseof complex Gaussian coefficients, we use the work of Peres and Vir{\'a}g [PV05]to obtain explicit formulas for the limiting objects.

The largest root of random Kac polynomials is heavy tailed