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On the determinant in Bray-Moore's TAP complexity formula

David Belius, Francesco Concetti, Giuseppe Genovese

TL;DR

This work resolves the determinant computation underlying TAP complexity for the Sherrington–Kirkpatrick landscape by rigorously validating Bray–Moore's exponential-scale formula for the random Hessian determinant and by deriving a subleading prefactor from an isolated outlier eigenvalue. The authors develop a robust framework around GOE matrices perturbed by a diagonal (and, later, low-rank) term, using Stieltjes transforms and the additive convolution with the semicircle law to connect determinant asymptotics to $\int \log|x|\,\mu_m(dx)$. A key advance is handling large operator-norm perturbations and the presence of a near-zero outlier, enabling precise asymptotics and a correction term that becomes relevant near $m\approx0$, particularly when $h=0$. The results bridge rigorous random-matrix theory with TAP-level complexity calculations, improving the mathematical grounding for exponential growth rates of metastable states in spin-glass models and informing future rigorous complexity analyses of TAP solutions.

Abstract

In the computation of the TAP complexity, originally carried out by Bray and Moore, a fundamental step is to calculate the determinant of a random Hessian. As the replica method does not give a clear prescription, physicists debated how to perform this computation and its consequences on the TAP complexity for a long time. In this paper we prove the original Bray and Moore formula for the behaviour of the determinant at exponential scale to be correct, and compute an important prefactor coming from a small outlier in the spectrum.

On the determinant in Bray-Moore's TAP complexity formula

TL;DR

This work resolves the determinant computation underlying TAP complexity for the Sherrington–Kirkpatrick landscape by rigorously validating Bray–Moore's exponential-scale formula for the random Hessian determinant and by deriving a subleading prefactor from an isolated outlier eigenvalue. The authors develop a robust framework around GOE matrices perturbed by a diagonal (and, later, low-rank) term, using Stieltjes transforms and the additive convolution with the semicircle law to connect determinant asymptotics to . A key advance is handling large operator-norm perturbations and the presence of a near-zero outlier, enabling precise asymptotics and a correction term that becomes relevant near , particularly when . The results bridge rigorous random-matrix theory with TAP-level complexity calculations, improving the mathematical grounding for exponential growth rates of metastable states in spin-glass models and informing future rigorous complexity analyses of TAP solutions.

Abstract

In the computation of the TAP complexity, originally carried out by Bray and Moore, a fundamental step is to calculate the determinant of a random Hessian. As the replica method does not give a clear prescription, physicists debated how to perform this computation and its consequences on the TAP complexity for a long time. In this paper we prove the original Bray and Moore formula for the behaviour of the determinant at exponential scale to be correct, and compute an important prefactor coming from a small outlier in the spectrum.
Paper Structure (4 sections, 10 theorems, 102 equations)

This paper contains 4 sections, 10 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Stieltjes transform and additive convolution
  4. Proof of Theorem \ref{['thm:GOE_plus_D_det']} without the lower rank term

Key Result

Theorem 1.1

For all $\alpha>0$ there exists a constant $c:=c(\alpha)$ such that for all $N\ge 1$ and all $m \in (-1,1)^N$ satisfying and (cf. the second $\log$ term in eq:MainDet), where $\Upsilon$ is as in eq:defUps and $v$ is as in eq:defu-v. Furthermore, for each $N\ge 1$ there are finitely many $m\in(-1,1)^N$ for which eq:v_cond does not hold, and for these $m$ the l.h.s. of eq:v_cond equals zero and $

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 8 more