Overview
Research investigates the characteristic polynomial associated with self-normalized random matrices. These matrices are characterized by rows that are independent and normalized to possess a unit $\mathrm{L}^2$ norm. The entries of these matrices, prior to normalization, are assumed to exhibit regularly varying tails, quantified by a tail index $\alpha$, which ranges from 0 to 2.
The study specifically focuses on the behavior of this characteristic polynomial outside the unit disk, where it is found to converge to a random analytic function denoted as $F_\alpha$. This function $F_\alpha$ is further identified as a multiplicative chaos, whose structure is described in terms of Poisson point processes.
A key finding is the observation of a transition between distinct universal regimes. The family of limiting functions $(F_\alpha)_{\alpha\in[0,2]}$ interpolates between Poisson multiplicative chaos at $\alpha=0$ and Gaussian multiplicative chaos at the boundary $\alpha=2$. This indicates that self-normalization provides a matrix model in which a continuous transition from Poissonian to Gaussian regimes for the limiting characteristic polynomial is observed as the tail index $\alpha$ is varied.
A similar transitional pattern is identified regarding the fluctuations of the traces of these self-normalized matrices.
Research Context
The study contributes to the understanding of random matrix theory, specifically concerning matrices that undergo self-normalization. The concept of regularly varying tails for matrix entries is central to this investigation, influencing the asymptotic behavior of the characteristic polynomial. The identified multiplicative chaos provides a framework for describing the random analytic function $F_\alpha$, linking it to established probabilistic structures such as Poisson point processes.
Approach
The research methodology involves studying the characteristic polynomial of self-normalized random matrices. The definition of these matrices specifies that their rows are independent and are normalized to have a unit $\mathrm{L}^2$ norm. The unnormalized entries are characterized by regularly varying tails, with the tail index $\alpha \in [0,2]$ being a critical parameter in the analysis.
The approach includes proving the convergence of the characteristic polynomial outside the unit disk to a random analytic function $F_\alpha$. This function is then identified and described as a multiplicative chaos expressed via Poisson point processes. The study also examines the interpolation properties of the family of limiting functions $(F_\alpha)_{\alpha\in[0,2]}$ to characterize the transition between different universal regimes. Furthermore, fluctuations of the traces of self-normalized matrices are analyzed to identify analogous transitions.
Findings
- Outside the unit disk, the characteristic polynomial of self-normalized random matrices converges to a random analytic function $F_\alpha$.
- The function $F_\alpha$ is identified as a multiplicative chaos, which is described in terms of Poisson point processes.
- The family of limiting functions $(F_\alpha)_{\alpha\in[0,2]}$ demonstrates an interpolation between two universal regimes: Poisson multiplicative chaos when $\alpha=0$ and Gaussian multiplicative chaos at the boundary $\alpha=2$.
- This interpolation signifies that self-normalization presents a matrix model where a transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial is observable as the tail index varies.
- A comparable transition is observed for the fluctuations of the traces of self-normalized matrices.
- As an application of these results, the spectral radius of self-normalized matrices is derived to be asymptotically bounded above by one in probability for any symmetric entry distribution.
Potential Applications
As an application of the established results, it is derived that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability. This holds true for any symmetric entry distribution.