arXiv subjects

Jiahe Shen

Publications and source records attributed to Jiahe Shen.

16 recordsLinked to original sources

arXiv2026

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

In this note, we study the distribution of the rational canonical form of a random matrix over the finite field $\mathbb{F}_p$, whose entries are independent and $ε$-balanced with $ε\in(0,1-1/p]$. We show that, as the matrix size tends to infinity, the statistics converge to independent Cohen-Lenstra distributions, demonstrating the universality of this asymptotic behavior. In particular, we recover, as a special case, the uniform setting proved by Fulman in his thesis in 1997. Our proof uses the fact that the rational canonical form data of $A_n$ and the $\mathbb{F}_p[t]$-module structure of the function field cokernel $\Cok(tI_n-A_n)$ determine each other uniquely. Consequently, our question can be reformulated, equivalently, as the asymptotic distribution problem for this cokernel, which has been established by Cheong-Yu (arXiv:2303.09125).

math.PR↗

arXiv2026

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 $α_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{nα_n}{\log n}>1, $$ 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 $d$.

math.PR↗

arXiv2026

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 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.

math.PR↗

arXiv2026

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 odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.

math.PR↗

arXiv2026

$p$-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 consequences: a $p$-adic projection theorem for the maps $π_θ(x,y)=x+θy$, together with the corresponding exceptional set estimate giving a $p$-adic analogue of Oberlin's projection question; and a discretized fractal sum-product estimate over $\mathbb{Q}_p$, showing that sufficiently non-concentrated subsets of $\mathbb{Z}_p^\times$ cannot have both small sum set and small product set. The proof follows the projection-theoretic and multiscale machinery developed in the Euclidean works of Orponen-Shmerkin (arXiv:2301.10199) and Ren-Wang (arXiv:2308.08819). The main task is to rebuild this machinery in the non-archimedean setting, and along the way we develop several new $p$-adic inputs needed to overcome the ultrametric features of the problem.

math.MG↗

arXiv2026

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

We study the distribution of eigenvalues of Haar-random matrices over $\mathbb{Z}_p$ among algebraic extensions of $\mathbb{Q}_p$. Our results give $p$-adic analogues of the real-eigenvalue counting results of Edelman-Kostlan-Shub for the real Ginibre ensemble, but with a different degree behavior: while real eigenvalues form only a vanishing proportion in the real Ginibre ensemble, $p$-adic eigenvalues are asymptotically evenly distributed among possible extension degrees. We also show that the maximal unramified extension $\mathbb{Q}_p^{\mathrm{un}}$ captures all but a bounded expected number of eigenvalues, and that the expected number of eigenvalues outside $\mathbb{Q}_p^{\mathrm{un}}$ has a finite positive limit with an explicit upper bound. The proof uses correlation function formulas from the author's previous joint work with Van Peski (arXiv:2601.06283), together with uniform estimates over varying finite extensions. We also prove analogous results for roots of random Haar polynomials over $\mathbb{Z}_p$, using the correlation function formulas of Caruso (arXiv:2110.03942). These polynomial results are $p$-adic analogues of the real-root counting results of Edelman-Kostlan, again with behavior different from the real setting.

math.NT↗

arXiv2026

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

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 Bernoulli-Laplace model introduced by Scarabotti. Writing $n=ml$, where there are $m$ card types and each type appears $l$ times, we determine the limiting profile for the total variation distance to stationarity at times $t=\frac{n}{2}\left(\log n-\frac{1}{2}\log l+c\right)$, under the assumption $l=ω(1)$. Scarabotti previously established that this process exhibits cutoff at time $\frac{n}{2}(\log n-\frac{1}{2}\log l)$; our result refines this by identifying the precise asymptotic shape of convergence inside the cutoff window. We show that the limiting profile is asymptotically Gaussian, with different explicit forms in the regimes $m$ fixed and $m=ω(1)$. Together with our previous work on the fixed-$l$ regime, where the limiting profile is of Poisson type, this yields the cutoff profile for the random transposition shuffle on $n=ml$ repeated cards for the full range of parameters $m$ and $l$. Our argument has two main steps. First, we combine Scarabotti's Fourier-analytic framework for the many-urn Bernoulli-Laplace model with the approximation method of Jain-Sawhney (arXiv:2410.23944). More precisely, we compare the original shuffling measure with an explicitly tractable auxiliary measure directly on the repeated card quotient space, rather than passing through an intermediate comparison on the full symmetric group; this step relies in particular on our new estimates for Kostka numbers. Second, we reduce the limiting-profile problem to quotient fixed-point statistics and analyze them via Hoeffding-type combinatorial central limit theorems.

math.PR↗

arXiv2026

The $k$-cycle shuffling with repeated cards

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 $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle, including cutoff and explicit limiting profiles. For fixed $l$, we show that the walk exhibits cutoff at time $\frac{n}{k}\log n$ with window of order $\frac{n}{k}$, and we identify the limiting profile in terms of the total variation distance between Poisson distributions arising from quotient fixed-point statistics. When $l\to\infty$ with sufficiently slow growth, more precisely when $l=o(\log n)$, we prove that the cutoff location shifts to $\frac{n}{k}\left(\log n-\frac 12\log l\right)$, again with window of order $\frac{n}{k}$, and that the limiting profile is asymptotically Gaussian, arising from a Poisson comparison after normal approximation. The proof is based on an approximation of the shuffling measure by an explicitly tractable auxiliary measure, generalizing the $k=2$ case from Jain and Sawhney (arXiv:2410.23944). The representation-theoretic framework underlying the analysis of this auxiliary measure follows from the work of Hough (arXiv:1605.00911) and Nestoridi and Olesker-Taylor (arXiv:2005.13437)

math.PR↗

arXiv2026

Quantative universality for cokernels of matrices with symmetries

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:1402.5149). In the symmetric case, we reprove Hodges' universality theorem (arXiv:2311.07078), i.e. the version incorporating the canonical pairing from Wood's setting, and in the alternating case we reprove the local universality theorem of Nguyen-Wood (arXiv:2210.08526). A key advantage of our method is that it is quantitative: we obtain explicit error bounds, which are exponentially small in most regimes, thereby addressing Wood's question on effective convergence rates. Our argument is inspired by Maples' exposure-process and coupling viewpoint (arXiv:1301.1239) and uses a generalized form of Fourier-analytic estimates in the exponentially sharp style of Ferber-Jain-Sah-Sawhney (arXiv:2106.04049).

math.PR↗

arXiv2026

Groups with pairings, Hall modules, and Hall-Littlewood polynomials

We relate the combinatorics of Hall-Littlewood polynomials to that of abelian $p$-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian $p$-groups (without pairings) and Hall-Littlewood polynomials. Specifically, we define a module over the classical Hall algebra with basis indexed by groups with pairings, and explicitly relate its structure constants to Hall-Littlewood polynomials at different values of the parameter $t$. We also show certain expectation formulas with respect to Cohen-Lenstra type measures on groups with pairings. In the alternating case this gives a new and simpler proof of previous results of Delaunay-Jouhet.

math.CO↗

arXiv2026

Eigenvalues of $p$-adic random matrices

We develop the basic theory of eigenvalues of $p$-adic random matrices, analogous to the classical theory for random matrices over $\mathbb{R}$ and $\mathbb{C}$. Such eigenvalue statistics were proposed as a model for the zeroes of $p$-adic $L$-functions by Ellenberg-Jain-Venkatesh, who computed the limiting distribution of the number of eigenvalues in a unit disc. We compute the full joint distribution of the $n$ eigenvalues of an $n \times n$ matrix with Haar distribution, obtaining Coulomb gas type formulas as in the archimedean case, with Vandermonde terms leading to eigenvalue repulsion. From these Coulomb gas density functions we derive asymptotics of eigenvalue statistics as $n \to \infty$. These include exact computations, such as a closed form $$ρ(x,y) = 1 - θ_3(-\sqrt{p};||x-y||^2/p)$$ for the limiting pair correlation of eigenvalues in $\mathbb{Z}_p$, and similar results in quadratic extensions. Such formulas yield concrete numerical predictions on zeroes of $p$-adic $L$-functions. For eigenvalues in arbitrary extensions of $\mathbb{Q}_p$ we also give precise estimates on their pair-repulsion and expected number of eigenvalues in each extension. Finally, we compute the asymptotic probability that all eigenvalues lie in $\mathbb{Z}_p$. Our proofs combine results from several distinct areas: $p$-adic orbital integrals, roots of random $p$-adic polynomials, the Sawin-Wood moment method for random modules, and Markov chains associated with measures on integer partitions.

math.NT↗

arXiv2025

High-low method and $p$-adic Furstenberg set over the plane

We establish a $p$-adic analogue of a recent significant result of Ren-Wang (arXiv:2308.08819) on Furstenberg sets in the Euclidean plane. Building on the $p$-adic version of the high-low method from Chu (arXiv:2510.20104), we analyze cube-tube incidences in $\mathbb{Q}_p^2$ and prove that for $s < t < 2 - s$, any semi-well-spaced $(s,t)$-Furstenberg set over $\mathbb{Q}_p^2$ has Hausdorff dimension $\ge\frac{3s+t}{2}$. Moreover, as a byproduct of our argument, we obtain the sharp lower bounds $s+t$ (for $0<t\le s\le 1$) and $s+1$ (for $s+t\ge 2$) for general $(s,t)$-Furstenberg sets without the semi-well-spaced assumption, thereby confirming that all three lower bounds match those in the Euclidean case.

math.FA↗

arXiv2025

The Szemerédi-Trotter theorem over arbitrary field of characteristic zero

Let $\mathcal{P}$ be a set of $m$ points and $\mathcal{L}$ a set of $n$ lines in $K^2$, where $K$ is a field with char$(K)=0$. We prove the incidence bound $$\mathcal{I}(\mathcal{P},\mathcal{L})=O(m^{2/3}n^{2/3}+m+n).$$ Moreover, this bound is sharp and cannot be improved. This resolves the Szemerédi-Trotter incidence problem for arbitrary field of characteristic zero. The key tool of our proof is the Baby Lefschetz principle, which allows us to reduce the problem to the complex case. Based on this observation, we further derive several related results over $K$, including Beck's theorem, the Erdős-Szemerédi sum-product estimate, and incidence theorems involving more general algebraic objects.

math.CO↗

arXiv2025

Non-Archimedean GUE corners and Hecke modules

We compute the joint distribution of singular numbers for all principal corners of a $p$-adic Hermitian (resp. alternating) matrix with additive Haar distribution, the non-archimedean analogue of the GUE (resp. aGUE) corners process. In the alternating case we find that it is a Hall-Littlewood process, explaining -- and recovering as a corollary -- results of Fulman-Kaplan. In the Hermitian case we obtain a `marginal distribution' of a formal Hall-Littlewood process with both positive and negative transition `probabilities'. The proofs relate natural random matrix operations to structural results of Hironaka and Hironaka-Sato on modules over the spherical Hecke algebra, yielding other probabilistic statements of independent interest along the way.

math.PR↗

arXiv2025

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

We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean fields are fundamentally determined by Hall-Littlewood polynomials. Through applications of the Satake isomorphism, we extend Van Peski's results (arXiv:2011.09356, Theorem 1.3) to encompass arbitrary root systems. Leveraging this theoretical foundation, we further develop Shen's work (arXiv:2411.01104, Theorem 1.1) to demonstrate that both singular numbers and corners of such products exhibit minimal separation. This characterization enables the derivation of asymptotic properties for singular numbers in matrix products, particularly establishing the strong law of large numbers and central limit theorem for these quantities. Our results provide a unified framework connecting algebraic decomposition structures with probabilistic limit theorems in non-archimedean settings.

math.PR↗

arXiv2024

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

We establish the universality of the singular numbers in random matrix products over $\mathrm{GL}_n(\mathbb{Q}_p)$ as the number of products approaches infinity, with a fixed $n\ge 1$. We demonstrate that, under a broad class of distributions, which we term as ``split", the asymptotics of matrix products align with the sum of the singular numbers of the matrix corners. Specifically, when matrices are independent and identically distributed, we derive the strong law of large numbers and the central limit theorem. Our approach is inspired by Van Peski's work (arXiv:2011.09356), which examines products of $n\times n$ corners of Haar-distributed elements $\mathrm{GL}_N(\mathbb{Z}_p)$. We extend the method so that the criterion now works as long as the measures are left- and right-invariant under the multiplication of $\mathrm{GL}_n(\mathbb{Z}_p)$. Building on this approach for $\mathrm{GL}_n$, we further demonstrate similar universality for $\mathrm{GSp}_{2n}$ and discuss potential extensions to general split reductive groups.

math.PR↗