Author

Jiahe Shen

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Preprint Sep 2026

Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field

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 p...

Jiahe Shen · 0 citations

Preprint Jul 2026

$p$-adic Sum-Product, Projections, and Furstenberg Sets

Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$-adic plane $\mathbb{Q}_p^2$: every $(s,t)$-Furstenberg set $E\subset\mathbb{Q}_p^2$ satisfies $$ \dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}. $$ This matches the sharp lower bound in the Euclidean plane. We also derive two related...

Jiahe Shen · 0 citations

Preprint Aug 2026

The Resultant Distribution Method: Universality for $p$-adic Random Matrices and Polynomials

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 H...

Jiahe Shen · 0 citations

Preprint Jul 2026

Hitting time mixing for random $k$-cycles

In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For o...

Chen Shang, Jiahe Shen, Jiyue Zeng et al. · 1 citation