We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $\alpha \in [0,2]$. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function $F_\alpha$. We identify $F_\alpha$ as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions $(F_\alpha)_{\alpha\in[0,2]}$ interpolates between two universal regimes: Poisson multiplicative chaos at $\alpha=0$ and Gaussian multiplicative chaos at the boundary $\alpha=2$. 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.
We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, mult...
Yin-Qi Cai, Bo-Nan Chen, Xiang Fang et al.· 0 citations
Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $\mu_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in(0,1)$, we prove that, as $n \to \infty$, the empirical zero measure of the $k_n$...
We initiate the study of approximating the top eigenvalue and eigenvector of a random symmetric matrix $ A \in \mathbb{R}^{n\times n} $ using $ q(A)b $ where $q$ is a degree-$d$ polynomial and $b$ is a standard Gaussian vector independent of $A$. For spiked GOE $ Y = \lambda vv^\top + X $, we identify $ d_\star = \frac...
For fixed $k\ge3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map $\Theta_N:\Lambda^k\mathbb {F}_q^N\to\mathbb {F}_q^m$. At the exact balance $m\binom rk=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\operatorname{Gr}(r,\mathbb {F}_q...
The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In...
M. Bertucci, Sam Van Blarcom, Benjamin Glancy et al.· 0 citations
We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<\beta<-\frac12$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the...
Davrbek Oltiboev, K. Rakhimov, Marks Ruziboev· 0 citations
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