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arxiv: 2408.09256 · v3 · submitted 2024-08-17 · 🧮 math.PR

Large deviations for the smallest eigenvalue of a deformed GOE with an outlier

Jeanne Boursier , Alice Guionnet This is my paper

Pith reviewed 2026-05-23 21:59 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviationssmallest eigenvalueGOEoutlierdeformed ensemblerandom matricesspectral statistics
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The pith

A large deviation principle holds for the smallest eigenvalue of a GOE matrix plus a diagonal outlier.

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

The paper proves a large deviation principle for the smallest eigenvalue of the sum of a GOE matrix and a diagonal matrix that contains one outlier entry. This extends and unifies earlier large-deviation results that covered only special placements of the outlier or particular scaling regimes. A reader would care because the principle supplies an explicit rate function that governs how rare it is for the edge eigenvalue to sit far from its typical location in high-dimensional random matrices. The result applies whenever the outlier sits outside the bulk spectrum, recovering prior cases as instances of the same statement.

Core claim

We establish a large deviation principle for the smallest eigenvalue of a random matrix model composed of the sum of a GOE matrix and a diagonal matrix with an outlier. Our result generalizes and unifies previously studied cases.

What carries the argument

The large deviation principle for the smallest eigenvalue of the deformed GOE matrix with one separated outlier.

If this is right

  • Probabilities of atypical locations for the smallest eigenvalue decay exponentially in matrix dimension at a rate given by an explicit rate function.
  • All earlier large-deviation statements for the edge in this model class become special cases of the same principle.
  • The rate function remains well-defined under the separation condition on the outlier.
  • The same framework applies uniformly across the range of outlier strengths that satisfy the separation hypothesis.

Where Pith is reading between the lines

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

  • The result supplies a template that can be tested on the largest eigenvalue or on the bulk eigenvalues of the same deformed model.
  • It opens a route to large-deviation statements for the condition number or for the gap between the smallest eigenvalue and the bulk edge.

Load-bearing premise

The outlier lies sufficiently far from the bulk spectrum of the GOE matrix.

What would settle it

Numerical computation, for increasing matrix size N, of the empirical log-probability that the smallest eigenvalue exceeds a fixed threshold above its typical value, checked against linear scaling in N with the predicted rate function.

Figures

Figures reproduced from arXiv: 2408.09256 by Alice Guionnet, Jeanne Boursier.

Figure 1
Figure 1. Figure 1: Graph of Hν,t : x ∈ (−∞, ℓν) 7→ x + tGν(x). Figures A and B correspond to the first case, Figure C to the second case of (1.7) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph of the function x ∈ (−∞, ℓν) 7→ x + tGν(x) [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
read the original abstract

We establish a large deviation principle for the smallest eigenvalue of a random matrix model composed of the sum of a GOE matrix and a diagonal matrix with an outlier. Our result generalizes and unifies previously studied cases.

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

0 major / 3 minor

Summary. The manuscript establishes a large deviation principle for the smallest eigenvalue of the sum of a standard GOE matrix and a diagonal matrix containing a single outlier eigenvalue separated from the semicircular bulk. The result is claimed to generalize and unify several previously studied regimes for extreme eigenvalues in deformed GOE ensembles.

Significance. If the LDP and its rate function are correctly derived, the work supplies a unified large-deviation description for the smallest eigenvalue across a range of outlier strengths, extending existing results on deformed GOE matrices. The use of standard large-deviation techniques for the empirical spectral measure combined with direct analysis of the edge eigenvalue is a natural approach in this area.

minor comments (3)
  1. The normalization of the GOE matrix (variance of off-diagonal entries) and the precise location of the outlier relative to the semicircle edge should be stated explicitly in the introduction and in the statement of the main theorem to facilitate comparison with prior works.
  2. Notation for the rate function I(·) and the limiting position of the outlier should be introduced once and used consistently; currently the abstract and introduction employ slightly different symbols for the same quantities.
  3. The proof sketch in §3 relies on standard LDP for the empirical measure plus a separate argument for the smallest eigenvalue; a short remark clarifying why the usual variational problem does not directly yield the edge LDP would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments are provided in the report, so we have no specific points requiring response or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an LDP theorem for the smallest eigenvalue via standard large-deviation analysis of the empirical spectral measure combined with direct control on the outlier eigenvalue. All definitions (GOE normalization, outlier separation from the semicircle law) are given explicitly as inputs, and the rate function is derived from the variational problem without reducing to a fitted parameter or self-referential definition. The generalization of prior cases rests on independent technical estimates rather than any self-citation chain or renaming of known results. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5545 in / 887 out tokens · 33732 ms · 2026-05-23T21:59:03.207413+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

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