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Diffstat (limited to 'when_annealed.tex')
| -rw-r--r-- | when_annealed.tex | 20 |
1 files changed, 11 insertions, 9 deletions
diff --git a/when_annealed.tex b/when_annealed.tex index dd57ec2..469fd6e 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -56,12 +56,14 @@ Random high-dimensional energies, cost functions, and interaction networks are important in many fields. The energy landscape of glasses, the likelihood landscape of machine learning and inference, and the interactions between -organisms in an ecosystem are just a few examples. A traditional tool for +organisms in an ecosystem are just a few examples \cite{Stein_1995_Broken, Krzakala_2007_Landscape, Altieri_2021_Properties, Yang_2023_Stochastic}. A traditional tool for making sense of their behavior is to analyze the statistics of points where -their dynamics are stationary. For energy or cost landscapes, these correspond -to the minima, maxima, and saddles, while for ecosystems and other non-gradient -dynamical systems these correspond to equilibria of the dynamics. When many -stationary points are present, the system is considered complex. +their dynamics are stationary \cite{Cavagna_1998_Stationary, +Fyodorov_2004_Complexity, Fyodorov_2007_Density, Bray_2007_Statistics}. For +energy or cost landscapes, these correspond to the minima, maxima, and saddles, +while for ecosystems and other non-gradient dynamical systems these correspond +to equilibria of the dynamics. When many stationary points are present, the +system is considered complex. Despite the importance of stationary point statistics for understanding complex behavior, they are often calculated using an uncontrolled approximation. @@ -69,12 +71,12 @@ Because their number is so large, it cannot be reliably averaged. The annealed approximation takes this average anyway, risking a systematic bias by rare and atypical samples. The annealed approximation is known to be exact for certain models and in certain circumstances, but it is used outside those circumstances -without much reflection \cite{Wainrib_2013_Topological, +without much reflection \cite{Wainrib_2013_Topological, Kent-Dobias_2021_Complex, Gershenzon_2023_On-Site}. In a few cases researches have made instead the better-controlled quenched average, which averages the logarithm of the number of stationary points, and find deviations from the annealed approximation with important implications for the system's behavior \cite{Muller_2006_Marginal, -Ros_2019_Complexity, Kent-Dobias_2023_How, Ros_2023_Quenched}. Generically, +Ros_2019_Complex, Kent-Dobias_2023_How, Ros_2023_Quenched}. Generically, the annealed approximation to the complexity is wrong when a nonvanishing fraction of pairs of stationary points have nontrivial correlations in their mutual position. @@ -386,8 +388,8 @@ proportional to -2(f''-f')u_fw_f -2\log^2\frac{f''}{f'}f'^2f''v_f \end{equation} -If $G_f>0$, then the bifurcating solutions exist, and there is someplace where -the annealed solution is corrected by a {\oldstylenums1\textsc{rsb}} solution. +If $G_f>0$, then the bifurcating solutions exist, and there are some saddles whose +complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution. Therefore, $G_f>0$ is a condition to see {\oldstylenums1}\textsc{rsb} in the complexity. If $G_f<0$, then there is nowhere along the extremal line where saddles can be described by such a complexity. The range of $3+s$ models where |