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arxiv: 2605.20444 · v1 · pith:KMHM66O5new · submitted 2026-05-19 · 🧮 math.NT · math.PR

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

Jiahe Shen This is my paper

Pith reviewed 2026-05-21 06:45 UTC · model grok-4.3

classification 🧮 math.NT math.PR
keywords p-adic random matriceseigenvalue distributionalgebraic extensionsunramified extensionrandom polynomialsHaar measurecorrelation functionsp-adic numbers
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The pith

p-adic random matrix eigenvalues are asymptotically evenly distributed among algebraic extension degrees of Q_p

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the distribution of eigenvalues of Haar-random matrices with entries in the p-adic integers across algebraic extensions of Q_p. It shows that these eigenvalues spread asymptotically evenly among possible extension degrees, in contrast to the real Ginibre ensemble where real eigenvalues form a vanishing proportion. The maximal unramified extension Q_p^un accounts for all but a bounded expected number of eigenvalues, while the expected number outside it converges to a finite positive limit with an explicit upper bound. Parallel results hold for roots of random Haar polynomials over Z_p, again using correlation functions but yielding different degree behavior from the real case.

Core claim

For a Haar-random matrix over Z_p the eigenvalues are asymptotically evenly distributed among the possible degrees of algebraic extensions of Q_p. The maximal unramified extension Q_p^un captures all but a bounded expected number of these eigenvalues as the matrix size grows, and the expected number lying outside Q_p^un has a finite positive limit that admits an explicit upper bound. The same distribution statements hold for the roots of Haar-random polynomials over Z_p.

What carries the argument

Correlation function formulas for joint eigenvalue counts, extended uniformly over all finite extensions of Q_p

If this is right

  • The expected number of eigenvalues outside the maximal unramified extension stays bounded for large matrices.
  • Eigenvalues appear with positive asymptotic density in extensions of every fixed degree.
  • Random p-adic polynomials exhibit the same even distribution of roots across extension degrees.
  • An explicit upper bound is available for the expected count of eigenvalues requiring ramified extensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical sampling of moderate-sized matrices for small p could directly verify the finite limit outside Q_p^un.
  • The uniform spread may simplify sampling or approximation schemes by focusing primarily on unramified extensions.
  • The contrast with real-eigenvalue vanishing suggests p-adic models capture arithmetic statistics more uniformly than archimedean ones.

Load-bearing premise

The correlation function formulas extend uniformly to all finite extensions of Q_p without additional error terms that would alter the leading asymptotics.

What would settle it

A computation showing that the expected number of eigenvalues in ramified extensions grows unbounded with matrix size would falsify the bounded-expectation claim.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the distribution of eigenvalues of Haar-random matrices over Z_p among algebraic extensions of Q_p. It claims that these eigenvalues are asymptotically evenly distributed among possible extension degrees (in contrast to the vanishing proportion of real eigenvalues in the Ginibre ensemble), that the maximal unramified extension Q_p^un captures all but a bounded expected number of eigenvalues, and that the expected number outside Q_p^un has a finite positive limit with an explicit upper bound. Analogous results are proved for roots of random Haar polynomials over Z_p, relying on correlation-function formulas from prior works.

Significance. If the results hold, they provide p-adic analogues to the Edelman-Kostlan-Shub and Edelman-Kostlan counting theorems with qualitatively different behavior driven by ramification. The approach of importing correlation identities from arXiv:2601.06283 and arXiv:2110.03942 and supplementing them with uniform estimates over extensions is a clear strength when the uniformity is controlled at leading order; such results would be of interest to researchers in p-adic random matrix theory and arithmetic statistics.

major comments (2)
  1. [Section on uniform estimates over extensions] The central claims on even distribution by degree and the finite positive limit outside Q_p^un rest on applying the k-point correlation functions uniformly over all finite extensions K/Q_p. The manuscript invokes 'uniform estimates' but does not supply explicit error-term bounds showing that the remainder in the correlation functions is o(1) as [K:Q_p] or the ramification index tends to infinity in the scaling regime used for the expected-count integrals (see the derivation leading to the limit statement).
  2. [§4] §4 (or the section deriving the expected count outside Q_p^un): the passage from the correlation formulas of arXiv:2601.06283 to the claimed finite positive limit with explicit upper bound requires that any degree-dependent error remain negligible after integration against the appropriate measure. Without a quantitative statement that the error is smaller than the main term uniformly in the relevant range, the leading asymptotic could be altered.
minor comments (2)
  1. [Abstract] The abstract states the main results clearly but could indicate in one sentence the form of the uniform estimates (e.g., whether they are O(1/[K:Q_p]) or similar) to help readers assess the scope immediately.
  2. [Introduction] Notation for ramification index and residue degree is introduced late; moving a short paragraph on local field notation to the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance as a p-adic analogue of classical results. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Section on uniform estimates over extensions] The central claims on even distribution by degree and the finite positive limit outside Q_p^un rest on applying the k-point correlation functions uniformly over all finite extensions K/Q_p. The manuscript invokes 'uniform estimates' but does not supply explicit error-term bounds showing that the remainder in the correlation functions is o(1) as [K:Q_p] or the ramification index tends to infinity in the scaling regime used for the expected-count integrals (see the derivation leading to the limit statement).

    Authors: We appreciate this precise observation. The uniform estimates invoked in the manuscript are obtained by combining the correlation-function identities of arXiv:2601.06283 with bounds that are uniform in the extension degree and ramification index; these suffice to control the leading-order terms in the integrals that produce the even distribution and the finite limit. Nevertheless, we agree that the absence of fully explicit remainder bounds makes the argument less transparent. In the revised version we will insert a new lemma (or subsection) that supplies quantitative error estimates, showing that the remainders are o(1) uniformly in the scaling regime relevant to the expected-count integrals. This addition will not alter any of the stated results. revision: yes

  2. Referee: [§4] §4 (or the section deriving the expected count outside Q_p^un): the passage from the correlation formulas of arXiv:2601.06283 to the claimed finite positive limit with explicit upper bound requires that any degree-dependent error remain negligible after integration against the appropriate measure. Without a quantitative statement that the error is smaller than the main term uniformly in the relevant range, the leading asymptotic could be altered.

    Authors: We acknowledge the referee’s concern about the integration step. The argument in §4 proceeds by integrating the correlation functions against a measure supported on extensions of increasing degree; the uniformity already ensures that degree-dependent errors integrate to a quantity that is bounded independently of the cutoff. To make this step fully rigorous, the revised manuscript will include an explicit estimate showing that the integrated error is at most a constant (independent of the degree cutoff) times the main term, which is negligible in the limit. This quantitative control will confirm that the finite positive limit and the explicit upper bound remain unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on prior independent formulas plus new uniform estimates

full rationale

The paper states that its 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' and analogously invokes Caruso (arXiv:2110.03942) for the polynomial case. These prior results are separate preprints whose validity is independent of the present work. The current manuscript contributes the uniform estimates needed to extend the correlation formulas across all finite extensions K/Q_p, which directly supports the leading asymptotics for eigenvalue distribution among degrees and the bounded expectation outside Q_p^un. No equation or claim reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence and uniformity of correlation functions for the p-adic Ginibre and polynomial ensembles; these are imported from prior works rather than re-derived. No new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Haar measure on Z_p induces a well-defined probability measure on the space of n-by-n matrices over Z_p for each fixed n.
    Invoked implicitly when speaking of Haar-random matrices; standard in p-adic random matrix theory but required for the ensemble to be defined.
  • domain assumption Correlation functions derived in arXiv:2601.06283 and arXiv:2110.03942 remain valid and admit uniform bounds when the base field is replaced by an arbitrary finite extension of Q_p.
    This uniformity is the key technical step that converts the correlation formulas into degree-distribution statements; it is stated as an estimate obtained in the present work.

pith-pipeline@v0.9.0 · 5765 in / 1704 out tokens · 27444 ms · 2026-05-21T06:45:59.422111+00:00 · methodology

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