Characteristic polynomial of self-normalized random matrices
Abstract
We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit L2 norm. The entries before normalization are assumed to have regularly varying tails with tail index α∈ [0,2]. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function Fα. We identify Fα as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions (Fα)α∈[0,2] interpolates between two universal regimes: Poisson multiplicative chaos at α=0 and Gaussian multiplicative chaos at the boundary α=2. Thus, self-normalization provides a matrix model where one observes the transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial as the tail index varies. A similar transition is found for the fluctuations of the traces of self-normalized matrices. As an application of our results, we derive that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability for any symmetric entry distribution.
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BibTeX
@misc{francois2026characteristic,
title = {{Characteristic polynomial of self-normalized random matrices}},
author = {Quentin François and Johannes Heiny and Xuechun Hu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28169},
eprint = {2608.28169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28169},
}
RIS
TY - UNPB
TI - Characteristic polynomial of self-normalized random matrices
AU - Quentin François
AU - Johannes Heiny
AU - Xuechun Hu
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28169
UR - https://arxiv.org/abs/2608.28169
AB - We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit L^2 norm. The entries before normalization are assumed to have regularly varying tails with tail index α∈ [0,2]. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function F_α. We identify F_α as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions (F_α)_α∈[0,2] interpolates between two universal regimes: Poisson multiplicative chaos at α=0 and Gaussian multiplicative chaos at the boundary α=2. Thus, self-normalization provides a matrix model where one observes the transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial as the tail index varies. A similar transition is found for the fluctuations of the traces of self-normalized matrices. As an application of our results, we derive that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability for any symmetric entry distribution.
ER -