Jiahe Shen

activity

20242026

collaborators

12 papers

math.PR2026

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

Jiahe Shen

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_{n} \in Mat_{n} (F_{p})$ has independent and $α_{n}$ -balanced entries. We prov…

math.PR2026

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

Jiahe Shen

We prove universality of limiting local eigenvalue statistics for random matrices over $Z_{p}$ . In previous work of the author and Van Peski (arXiv:2601.06283), the limiting…

math.PR2026

Hitting time mixing for random $k$ -cycles

Chen Shang, Jiahe Shen, Jiyue Zeng +1

In this paper, we study the random walk on the symmetric group $S_{n}$ generated by the conjugacy class of $k$ -cycles, where $2 \leq k = o (n / (lo g n)^{4})$ . We prove that the wa…

math.MG2026

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

Jiahe Shen

Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$ -adic plane $Q_{p}^{2}$ : every $(s, t)$ -Furstenberg set $E \subset Q_{p}^{2}$ satisfies $$ \dim…

math.NT2026

Eigenvalue Distribution of $p$ -adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$ -adic Random Polynomials

Jiahe Shen

We study the distribution of eigenvalues of Haar-random matrices over $Z_{p}$ among algebraic extensions of $Q_{p}$ . Our results give $p$ -adic analogues of the real-e…

math.PR2026

The Cutoff Profile for Random Transpositions on Repeated Cards in the Full Range of Parameters

Jiahe Shen

The random transposition shuffle on repeated cards induces a Markov chain on the quotient space of arrangements with multiplicities, and is equivalent to the many-urn mean-field Be…

math.PR2026

The $k$ -cycle shuffling with repeated cards

Jiahe Shen

We investigate the $k$ -cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n = m l$ . We establish asymptot…

math.PR2026

Quantative universality for cokernels of matrices with symmetries

Jiahe Shen

We prove universality for cokernels of random integral matrices with symmetries via an approach different from the classical surjection moment method introduced by Wood (arXiv:1402…

math.CO2025

The Szemerédi-Trotter theorem over arbitrary field of characteristic zero

Jiahe Shen

Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in $K^{2}$ , where $K$ is a field with char $(K) = 0$ . We prove the incidence bound $$\mathcal{I}(\mathcal{P…

math.PR2025

Universality of rational canonical form for random matrices over a finite field

Jiahe Shen

In this note, we study the distribution of the rational canonical form of a random matrix over the finite field $F_{p}$ , whose entries are independent and $ε$ -balanced with…

math.PR2025

Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism

Jiahe Shen

We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean fields a…

math.PR2024

Gaussian universality of $p$ -adic random matrix products via corners

Jiahe Shen

We establish the universality of the singular numbers in random matrix products over $GL_{n} (Q_{p})$ as the number of products approaches infinity, with a fixed $n\ge…

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

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

Hitting time mixing for random k-cycles

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

Eigenvalue Distribution of p-adic Random Matrices Among Algebraic Extensions, with an Analogue for p-adic Random Polynomials

The Cutoff Profile for Random Transpositions on Repeated Cards in the Full Range of Parameters

The k-cycle shuffling with repeated cards

Quantative universality for cokernels of matrices with symmetries

The Szemerédi-Trotter theorem over arbitrary field of characteristic zero

Universality of rational canonical form for random matrices over a finite field

Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism

Gaussian universality of p-adic random matrix products via corners

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

Hitting time mixing for random $k$ -cycles

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

Eigenvalue Distribution of $p$ -adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$ -adic Random Polynomials

The $k$ -cycle shuffling with repeated cards

Gaussian universality of $p$ -adic random matrix products via corners