We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.
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.
Quentin François, J. Heiny, Xuechun Hu· 0 citations
We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.
We settle a conjecture of Bandeira, Kennedy and Singer, arising from their analysis of approximation ratios for the little Grothendieck problem over the unitary group, by proving that the average singular value of a normalized square complex Gaussian random matrix strictly decreases with the dimension. The starting point is a recurrence relation of Abreu, derived from the Christoffel--Darboux formula and a Tur\'{a}n determinant for Laguerre polynomials. This recurrence reduces the problem to the estimation of a mixed Laguerre integral. We transform this estimate into an explicit finite inequality by expanding the relevant integrals in the orthogonal basis associated with the weight $x^{1/2}\,\mathrm{e}^{-x}$. A telescoping identity then converts the resulting inequality into a positive form. The final positivity argument combines explicit estimates for central binomial coefficients, a logarithmic lower bound, and a finite verification of small dimensions. This completes the proof that the average singular value strictly decreases with the dimension in the square complex Gaussian case. As a consequence, the same monotonicity is obtained for $N\times(N+\lambda)$ complex Gaussian matrices with fixed rectangularity $\lambda=0,1,2,\dots$.
Let $d_n=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d_n$ independent uniformly random permutation matrices are linearly independent with probability $1-O(n^{-1/2})$. Conditioning on distinctness gives the same conclusion for a uniformly random $d_n$-element subset, thereby confirming a conjecture of Kushwaha and Tripathi. The proof combines three ingredients: a mod-$2$ complexity parameter for assignment functionals, the characteristic-function estimate of Roos in the form recorded by Do--Nguyen--Phan--Tran--Vu, and a kernel decomposition argument of Ferber--Kwan--Sauermann. For the uniform-subset model, we also record the elementary lower bound $\exp(3/2+o(1))n^2e^{-n}$ coming from an unoccupied matrix position.
We construct strongly convergent finite-dimensional random matrix models for finite $q$-Gaussian family in the range $\vert q \vert<\sqrt{2}-1$. The construction has two stages. First, we show that normalized sums of graph-product semicirculars over an Erdos-Renyi graph satisfy the $q$-Toeplitz relations up to an operator norm error converging to zero in probability. Using ultraproduct methods, this yields complete strong convergence, uniformly over all matrix coefficient dimensions, for noncommutative polynomials with bounded degree. Second, we use a quantitative tensor-GUE for graph-product semicirculars to convert these operator models into finite-dimensional random matrices. For every fixed polynomial degree, the convergence is uniform over coefficient dimensions that may be larger than the dimension of the random matrices. As applications, in the above range of $q$, the C$^\ast$-algebra generated by a finite $q$-Gaussian family is MF, and the Brown-Douglas-Fillmore extension semigroup of the nontrivial C$^\ast$-algebra generated by a finite $q$-Gaussian family is not a group.
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 this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.
M. Bertucci, Sam Van Blarcom, Benjamin Glancy et al.· 0 citations
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