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\title{
  When is the annealed complexity correct?
}

\author{Jaron Kent-Dobias}
\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma}

\maketitle
\begin{abstract}
  The difference between quenched and annealed averages is crucial for
  disordered systems. In isotropic mean-field systems, they differ when replica
  symmetry is broken. When computing the average free energy in equilibrium,
  there are robust conditions to understand when {\oldstylenums1}\textsc{rsb}
  is sufficient. When computing the average complexity, or the number of
  stationary points of the energy, there is only robust reasoning at the ground
  state, where a {\oldstylenums1}\textsc{rsb} equilibrium implies that the
  annealed complexity \emph{at the ground state} is correct. Here, we
  demonstrate that in the mixed spherical models, the annealed complexity can
  be wrong away from the ground state even when the equilibrium free energy is
  guaranteed to be at most {\oldstylenums1}\textsc{rsb} everywhere. Therefore,
  simple equilibrium order cannot be used to assume a simple landscape
  geometry.
\end{abstract}

Parisi construction, $f''(q)^{-1/2}$ is concave

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