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| -rw-r--r-- | frsb_kac-rice.bib | 55 | ||||
| -rw-r--r-- | frsb_kac-rice.tex | 6 |
2 files changed, 58 insertions, 3 deletions
diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index d308ddc..d656c30 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -152,6 +152,20 @@ doi = {10.1103/physrevlett.98.150201} } +@article{Cavagna_1997_An, + author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio}, + title = {An investigation of the hidden structure of states in a mean-field spin-glass model}, + journal = {Journal of Physics A: Mathematical and General}, + publisher = {IOP Publishing}, + year = {1997}, + month = {10}, + number = {20}, + volume = {30}, + pages = {7021--7038}, + url = {https://doi.org/10.1088%2F0305-4470%2F30%2F20%2F009}, + doi = {10.1088/0305-4470/30/20/009} +} + @article{Cavagna_1997_Structure, author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio}, title = {Structure of metastable states in spin glasses by means of a three replica potential}, @@ -166,6 +180,33 @@ doi = {10.1088/0305-4470/30/13/004} } +@article{Cavagna_1998_Stationary, + author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio}, + title = {Stationary points of the {Thouless}-{Anderson}-{Palmer} free energy}, + journal = {Physical Review B}, + publisher = {American Physical Society (APS)}, + year = {1998}, + month = {5}, + number = {18}, + volume = {57}, + pages = {11251--11257}, + url = {https://doi.org/10.1103%2Fphysrevb.57.11251}, + doi = {10.1103/physrevb.57.11251} +} + +@article{Cavagna_2005_Cavity, + author = {Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio}, + title = {Cavity method for supersymmetry-breaking spin glasses}, + journal = {Physical Review B}, + publisher = {American Physical Society (APS)}, + year = {2005}, + month = {1}, + number = {2}, + volume = {71}, + url = {https://doi.org/10.1103%2Fphysrevb.71.024422}, + doi = {10.1103/physrevb.71.024422} +} + @article{Charbonneau_2014_Fractal, author = {Charbonneau, Patrick and Kurchan, Jorge and Parisi, Giorgio and Urbani, Pierfrancesco and Zamponi, Francesco}, title = {Fractal free energy landscapes in structural glasses}, @@ -429,6 +470,20 @@ stochastic localization}, doi = {10.1103/physrevlett.120.085705} } +@inproceedings{Giardina_2005_Supersymmetry, + author = {Giardina, Irene and Cavagna, Andrea and Parisi, Giorgio}, + title = {Supersymmetry and metastability in disordered systems}, + publisher = {World Scientific}, + year = {2005}, + month = {9}, + pages = {204--209}, + url = {https://doi.org/10.1142%2F9789812701558_0023}, + doi = {10.1142/9789812701558_0023}, + booksubtitle = {Proceedings of the 31st Workshop of the International School of Solid State Physics, {Erice}, {Sicily}, {Italy}, 20 – 26 {July} 2004}, + booktitle = {Complexity, Metastability and Nonextensivity}, + editor = {Beck, C and Benedek, G and Rapisarda, A and Tsallis, C} +} + @article{Gross_1985_Mean-field, author = {Gross, D. J. and Kanter, I. and Sompolinsky, H.}, title = {Mean-field theory of the {Potts} glass}, 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 $D=\hat\beta R$ |