Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

@article{Fyodorov2019ManifoldsPB,
  title={Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum},
  author={Yan V. Fyodorov and Pierre le Doussal},
  journal={Journal of Statistical Physics},
  year={2019},
  volume={179},
  pages={176-215}
}
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 $$\mu $$ μ . We are interested in the mean spectral density $$\rho (\lambda )$$ ρ ( λ ) of the Hessian matrix $${{\mathcal {K}}}$$ K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for $$\rho (\lambda )$$ ρ ( λ ) for a fixed $$L^d$$ L d in the $$N \rightarrow… 
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