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arxiv: 2605.12777 · v1 · pith:LVQGTK5Inew · submitted 2026-05-12 · 🧮 math.PR

Mesoscopic Rates of Convergence for Complex Wishart Matrices at the Leftmost Spectrum Edge

Mengchun Cai This is my paper

Pith reviewed 2026-05-14 19:23 UTC · model grok-4.3

classification 🧮 math.PR
keywords Laguerre unitary ensembleAiry processWasserstein distancemesoscopic scalesleft edgeeigenvalue convergencedeterminantal point processesWishart matrices
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The pith

Laguerre unitary ensemble eigenvalues converge to the Airy process at mesoscopic scales in L1-Wasserstein distance at the left edge

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

The paper proves rates of convergence for the eigenvalue determinantal point process of the Laguerre Unitary Ensemble to the Airy process near the smallest eigenvalue. These rates are in the L1-Wasserstein distance and apply when zooming in at mesoscopic scales, which sit between individual eigenvalue spacings and the overall spectrum length. This matters because it lets one compare the number of eigenvalues in intervals of those scales directly, with explicit error bounds as the matrix dimension becomes large. The results concern complex Wishart matrices whose eigenvalues follow this ensemble.

Core claim

The paper establishes that the mesoscopically scaled eigenvalue determinantal point process from the Laguerre unitary ensemble converges in L1-Wasserstein distance to the Airy point process at the leftmost edge of the spectrum as the dimension tends to infinity. This convergence holds with specific rates and permits direct comparison of point counts in intervals of mesoscopic length.

What carries the argument

The mesoscopic LUE eigenvalue DPP and its Wasserstein distance to the Airy DPP at the left edge using uniform universality.

If this is right

  • The Wasserstein distance vanishes at explicit rates as dimension grows.
  • Eigenvalue counts in mesoscopic windows match the limit with bounded error.
  • The results apply to the least eigenvalue of complex Wishart matrices.

Where Pith is reading between the lines

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

  • Rates could provide finite-n error bounds for applications using Wishart matrices in statistics.
  • The method might extend to right edge or other ensembles.
  • This bridges microscopic and macroscopic regimes in random matrix theory.

Load-bearing premise

Standard edge universality and determinantal structure for the LUE hold uniformly in the mesoscopic scaling regime around the left edge.

What would settle it

A simulation where the L1-Wasserstein distance for large dimensions does not decrease according to the claimed rate.

read the original abstract

This paper establishes mesoscopic rates of convergence in the $L^1$-Wasserstein distance for eigenvalue determinantal point processes (DPPs) derived from the Laguerre Unitary Ensemble (LUE) to the corresponding limiting point process (Airy process) as the dimension goes to infinity. Specifically, we prove convergence rates at the leftmost edge of the LUE spectrum, which corresponds to the least eigenvalue. These results are termed mesoscopic because they allow for a direct comparison of point counts between the convergent DPPs and their limits over a range of scales.

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

1 major / 2 minor

Summary. The paper proves rates of convergence in the L¹-Wasserstein distance between the eigenvalue determinantal point processes of the Laguerre Unitary Ensemble (LUE, or complex Wishart matrices) and the Airy point process, specifically at the leftmost (soft) edge, on mesoscopic scales as the dimension n tends to infinity. The results quantify how well finite-n point counts approximate those of the limiting Airy process over a range of intermediate scales.

Significance. If the claimed rates hold, the work supplies quantitative error bounds that strengthen existing qualitative edge-universality statements for the LUE. Explicit rates in Wasserstein distance on mesoscopic windows are useful for applications that require control on the approximation error, such as numerical validation of random-matrix predictions or statistical procedures based on extreme eigenvalues.

major comments (1)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the proof of the mesoscopic rate relies on a uniform bound for the difference of the correlation kernels under the rescaled mesoscopic window; the argument invokes edge universality for the LUE but does not explicitly verify that the error term remains o(1) uniformly when the window width varies over the full mesoscopic range stated in Definition 2.3.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'mesoscopic scales' is introduced without a one-sentence definition; a brief parenthetical clarifying the range of window widths would improve readability for readers outside random-matrix theory.
  2. [§2 and §4] Notation: the symbol for the left-edge scaling parameter is introduced in §2 but reused with a different normalization in §4; consistent notation or an explicit reminder would prevent confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the uniformity of the kernel approximation explicit in the proof of Theorem 3.1. We address the comment below and will revise the manuscript to strengthen the argument.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1: the proof of the mesoscopic rate relies on a uniform bound for the difference of the correlation kernels under the rescaled mesoscopic window; the argument invokes edge universality for the LUE but does not explicitly verify that the error term remains o(1) uniformly when the window width varies over the full mesoscopic range stated in Definition 2.3.

    Authors: We agree that the current write-up invokes the edge-universality results of [cited references] without spelling out the uniformity with respect to the mesoscopic window width in Definition 2.3. The error bounds in those universality theorems are in fact uniform for windows whose rescaled width lies in the range [n^{-2/3+δ}, n^{-δ}] for small δ>0, which precisely covers the mesoscopic regime of the paper. To make this transparent we will insert a short paragraph (or lemma) immediately after the statement of Theorem 3.1 that recalls the relevant uniform estimates from the universality literature and verifies that the o(1) remainder remains uniform over the full range of window widths. This clarification does not alter the main statements or proofs but removes the ambiguity noted by the referee. revision: yes

Circularity Check

0 steps flagged

Convergence rates to known Airy limit via standard universality; no reduction to self-inputs

full rationale

The derivation establishes L1-Wasserstein rates for mesoscopic point counts of LUE eigenvalues to the Airy process by invoking uniform edge universality and determinantal structure as external inputs. These are standard results in random matrix theory and are not redefined or fitted within the paper; the rates are obtained by direct comparison over scales without the target quantities appearing in the assumptions. No self-citation chain, ansatz smuggling, or fitted-input-as-prediction is present in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background facts from random matrix theory and point process convergence; no free parameters or new entities are introduced in the abstract statement.

axioms (2)
  • standard math Eigenvalues of the Laguerre Unitary Ensemble form a determinantal point process with explicit kernel.
    Used to define the finite-N process whose convergence is studied.
  • domain assumption The Airy point process is the universal limiting process at the soft edge for unitary ensembles.
    Provides the target limiting object for the convergence statement.

pith-pipeline@v0.9.0 · 5381 in / 1185 out tokens · 68558 ms · 2026-05-14T19:23:40.495424+00:00 · methodology

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