Hessian eigenvalue distribution in a random Gaussian landscape

TL;DR

This paper investigates the eigenvalue distribution of the Hessian matrix in high-dimensional Gaussian landscapes, extending existing methods to include sub-leading effects and exploring implications for cosmological vacuum stability and inflation.

Contribution

It introduces an extended saddle point method and a new stochastic process approach to accurately determine the Hessian eigenvalue distribution, especially at the spectrum's edge.

Findings

Extended saddle point method incorporating sub-leading terms.

Equilibrium distribution derived via Dyson Brownian motion.

Consistent results between the two approaches in applicable cases.

Abstract

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of $1/N$ expansion, where $N$ is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.