# Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum

## TL;DR

This paper analyzes the spectral properties of the Hessian matrix at the energy minimum of a high-dimensional elastic manifold in a random landscape, revealing phase transitions and spectral gaps related to the confinement strength and replica symmetry breaking.

## Contribution

It extends previous zero-dimensional results to finite internal dimensions, deriving the spectral density equations and analyzing the transition between replica-symmetric and RSB phases in high-dimensional systems.

## Key findings

- For confinement curvature above a critical value, the Hessian spectrum is gapped.
- The spectral gap vanishes quadratically at the transition point.
- In the RSB phase, the spectrum can be gapless or have a zero edge depending on the RSB type.

## Abstract

We consider an elastic manifold of internal dimension $d$ and length $L$ pinned in a $N$ dimensional random potential and confined by an additional parabolic potential of curvature $\mu$. We are interested in the mean spectral density $\rho(\lambda)$ of the Hessian matrix $K$ at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for $\rho(\lambda)$ for a fixed $L^d$ in the $N \to \infty$ limit extending $d=0$ results of our previous work. A particular attention is devoted to analyzing the limit of extended lattice systems by letting $L\to \infty$. In all cases we show that for a confinement curvature $\mu$ exceeding a critical value $\mu_c$, the so-called "Larkin mass", the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at $\mu\to \mu_c$. For $\mu<\mu_c$ the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all $d$ as $(\mu_c-\mu)^4$ near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called "marginal cases" in $d=1,2$ which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.