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Jiahe Shen

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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 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.

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 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 $2\le k\le n-1$.

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