diff options
Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index fc1666b..03ec81a 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -52,7 +52,7 @@ been open ever since \cite{Parisi_1979_Infinite}. To date, the program of computing the number of stationary points---minima, saddle points, and maxima---of mean-field complex landscapes has been only carried out for a small subset of models, including most notably the (pure) $p$-spin model ($p>2$) -\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer} and for similar +\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary} and for similar energy functions inspired by molecular biology, evolution, and machine learning \cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. In a parallel development, it has evolved into an active field of probability @@ -609,8 +609,8 @@ The Kac--Rice problem has an approximate supersymmetry, which is found when the absolute value of the determinant is neglected and the trace of the Hessian is not fixed. This supersymmetry has been studied in great detail in the complexity of the Thouless--Anderson--Palmer (TAP) free energy -\cite{Annibale_2003_The, Annibale_2003_Supersymmetric, -Annibale_2004_Coexistence}. When the absolute value is dropped, the determinant in \eqref{eq:kac-rice} can be +\cite{Annibale_2003_The, ,Annibale_2003_Supersymmetric, +Annibale_2004_Coexistence, Cavagna_2005_Cavity, Giardina_2005_Supersymmetry}. When the absolute value is dropped, the determinant in \eqref{eq:kac-rice} can be represented by an integral over Grassmann variables, which yields a complexity depending on `bosons' and `fermions' that share the supersymmetry. The Ward identities associated with the supersymmetry imply that $D=\hat\beta R$ |