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

Jeanne Boursier, Alice Guionnet

TL;DR

This work establishes a sharp large deviation principle (LDP) at speed $N$ for the smallest eigenvalue of a deformed GOE, $X_N=G_N+B_N$, where $G_N$ is GOE with variance $t$ and $B_N$ is deterministic with spectral limit $\nu$ and possible outlier $\Lambda$. The authors develop a novel, fixed-point functional-equation approach that couples a projection onto the first eigenvector, Dirichlet fluctuations of mass on $B_N$’s eigenspaces, and the outlier dynamics via a function $\Phi(\Lambda,Y)$; this yields an explicit rate function $I_{\nu,t}^{\Lambda}$ expressed through free-convolution data $H_{\nu,t}$ and $S_{\nu}$. The main result unifies prior findings for the pure-outlier case (Maida 2007) and the no-outlier case (McKenna et al.), while extending to general $B_N$ through a sandwiching argument and continuity/convexity properties of the rate function. The analysis has implications for spin-glass TAP energy landscapes and BBP-type transitions, providing a principled probabilistic framework for the extreme-edge fluctuations of deformed random matrices with outliers.

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.

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

TL;DR

This work establishes a sharp large deviation principle (LDP) at speed for the smallest eigenvalue of a deformed GOE, , where is GOE with variance and is deterministic with spectral limit and possible outlier . The authors develop a novel, fixed-point functional-equation approach that couples a projection onto the first eigenvector, Dirichlet fluctuations of mass on ’s eigenspaces, and the outlier dynamics via a function ; this yields an explicit rate function expressed through free-convolution data and . The main result unifies prior findings for the pure-outlier case (Maida 2007) and the no-outlier case (McKenna et al.), while extending to general through a sandwiching argument and continuity/convexity properties of the rate function. The analysis has implications for spin-glass TAP energy landscapes and BBP-type transitions, providing a principled probabilistic framework for the extreme-edge fluctuations of deformed random matrices with outliers.

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.
Paper Structure (27 sections, 22 theorems, 296 equations, 3 figures)

This paper contains 27 sections, 22 theorems, 296 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main result
  4. Notation
  5. Heuristics
  6. Large deviations in the pure point case
  7. Projection onto the first eigenvector
  8. LDP for Dirichlet variables
  9. Outliers of $p_{v_1}^\perp B_N p_{v_1}^\perp$
  10. A functional inequality for the large deviation upper bound
  11. A functional inequality for the large deviation lower bound
  12. Solving the functional equation in the complement of the region $\Lambda<\omega_{\nu,t}(\lambda)$
  13. Solving the functional inequalities
  14. Step 1: study of the local minima such that $\Phi(\Lambda,\cdot)\ge \omega_{\nu,t}(\lambda)$.
  15. Step 2: study of the critical points such that $\Phi(\Lambda,\cdot)< \omega_{\nu,t}(\lambda)$ and $y_0>0$
  16. ...and 12 more sections

Key Result

Theorem 1.1

The smallest eigenvalue of $X_{N}=G_{N}+B_{N}$ converges almost surely towards $\ell_{\nu,t}^\Lambda$ when $N$ goes to infinity. Moreover the map $\Lambda\in(-\infty,\ell_{\nu})\mapsto \ell_{\nu,t}^{\Lambda}$ and $\nu\mapsto \ell_{\nu,t}^{\Lambda}$ are continuous (if $\mathcal{P}(\mathbb R)$ is endo

Figures (3)

  • Figure 1: Graph of $H_{\nu,t} :x\in (-\infty,\ell_\nu)\mapsto x+tG_\nu(x)$. Figures $A$ and $B$ correspond to the first case, Figure $C$ to the second case of \ref{['eq:defg']}.
  • Figure 2: The graph of the function $x\in (-\infty,\ell_\nu)\mapsto x+tG_\nu(x)$.
  • Figure :

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2: LDP for the smallest eigenvalue of Gaussian matrices
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1: LDP in the pure point case
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 32 more