Lots more references.

3 files changed, 169 insertions, 16 deletions

diff --git a/.gitignore b/.gitignore
index 6997128..9ab6f22 100644
--- a/.gitignore
+++ b/.gitignore
@@ -9,3 +9,4 @@
 *.out
 *.bcf
 *.run.xml
+*.synctex(busy)
diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index d337f2f..ecd2fed 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -1,3 +1,17 @@
+@article{Albert_2021_Searching,
+ author = {Albert, Samuel and Biroli, Giulio and Ladieu, François and Tourbot, Roland and Urbani, Pierfrancesco},
+ title = {Searching for the Gardner Transition in Glassy Glycerol},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2021},
+ month = {1},
+ number = {2},
+ volume = {126},
+ pages = {028001},
+ url = {https://doi.org/10.1103%2Fphysrevlett.126.028001},
+ doi = {10.1103/physrevlett.126.028001}
+}
+
 @article{Annibale_2003_Supersymmetric,
  author = {Annibale, Alessia and Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio},
  title = {Supersymmetric complexity in the {Sherrington}-{Kirkpatrick} model},
@@ -137,6 +151,20 @@
  doi = {10.1038/ncomms4725}
 }
 
+@article{Charbonneau_2015_Numerical,
+ author = {Charbonneau, Patrick and Jin, Yuliang and Parisi, Giorgio and Rainone, Corrado and Seoane, Beatriz and Zamponi, Francesco},
+ title = {Numerical detection of the {Gardner} transition in a mean-field glass former},
+ journal = {Physical Review E},
+ publisher = {American Physical Society (APS)},
+ year = {2015},
+ month = {7},
+ number = {1},
+ volume = {92},
+ pages = {012316},
+ url = {https://doi.org/10.1103%2Fphysreve.92.012316},
+ doi = {10.1103/physreve.92.012316}
+}
+
 @article{Crisanti_1992_The,
  author = {Crisanti, A. and Sommers, H.-J.},
  title = {The spherical $p$-spin interaction spin glass model: the statics},
@@ -221,6 +249,20 @@
  doi = {10.1103/physrevlett.71.173}
 }
 
+@article{Dennis_2020_Jamming,
+ author = {Dennis, R. C. and Corwin, E. I.},
+ title = {Jamming Energy Landscape is Hierarchical and Ultrametric},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2020},
+ month = {2},
+ number = {7},
+ volume = {124},
+ pages = {078002},
+ url = {https://doi.org/10.1103%2Fphysrevlett.124.078002},
+ doi = {10.1103/physrevlett.124.078002}
+}
+
 @article{ElAlaoui_2022_Sampling,
  author = {El Alaoui, Ahmed and Montanari, Andrea and Sellke, Mark},
  title = {Sampling from the {Sherrington}-{Kirkpatrick} {Gibbs} measure via algorithmic
@@ -303,6 +345,21 @@ stochastic localization},
  doi = {10.1016/0550-3213(85)90374-8}
 }
 
+@article{Geirhos_2018_Johari-Goldstein,
+ author = {Geirhos, K. and Lunkenheimer, P. and Loidl, A.},
+ title = {{Johari}-{Goldstein} Relaxation Far Below 
+{$T_g$}: Experimental Evidence for the {Gardner} Transition in Structural Glasses?},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2018},
+ month = {2},
+ number = {8},
+ volume = {120},
+ pages = {085705},
+ url = {https://doi.org/10.1103%2Fphysrevlett.120.085705},
+ doi = {10.1103/physrevlett.120.085705}
+}
+
 @article{Gross_1985_Mean-field,
  author = {Gross, D. J. and Kanter, I. and Sompolinsky, H.},
  title = {Mean-field theory of the {Potts} glass},
@@ -317,6 +374,34 @@ stochastic localization},
  doi = {10.1103/physrevlett.55.304}
 }
 
+@article{Hammond_2020_Experimental,
+ author = {Hammond, Andrew P. and Corwin, Eric I.},
+ title = {Experimental observation of the marginal glass phase in a colloidal glass},
+ journal = {Proceedings of the National Academy of Sciences},
+ publisher = {Proceedings of the National Academy of Sciences},
+ year = {2020},
+ month = {3},
+ number = {11},
+ volume = {117},
+ pages = {5714--5718},
+ url = {https://doi.org/10.1073%2Fpnas.1917283117},
+ doi = {10.1073/pnas.1917283117}
+}
+
+@article{Hicks_2018_Gardner,
+ author = {Hicks, C. L. and Wheatley, M. J. and Godfrey, M. J. and Moore, M. A.},
+ title = {{Gardner} Transition in Physical Dimensions},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2018},
+ month = {5},
+ number = {22},
+ volume = {120},
+ pages = {225501},
+ url = {https://doi.org/10.1103%2Fphysrevlett.120.225501},
+ doi = {10.1103/physrevlett.120.225501}
+}
+
 @article{Huang_2021_Tight,
  author = {Huang, Brice and Sellke, Mark},
  title = {Tight {Lipschitz} Hardness for Optimizing Mean Field Spin Glasses},
@@ -342,6 +427,34 @@ stochastic localization},
  url = {https://projecteuclid.org:443/euclid.bams/1183505112}
 }
 
+@article{Li_2021_Determining,
+ author = {Li, Huaping and Jin, Yuliang and Jiang, Ying and Chen, Jeff Z. Y.},
+ title = {Determining the nonequilibrium criticality of a {Gardner} transition via a hybrid study of molecular simulations and machine learning},
+ journal = {Proceedings of the National Academy of Sciences},
+ publisher = {Proceedings of the National Academy of Sciences},
+ year = {2021},
+ month = {3},
+ number = {11},
+ volume = {118},
+ pages = {e2017392118},
+ url = {https://doi.org/10.1073%2Fpnas.2017392118},
+ doi = {10.1073/pnas.2017392118}
+}
+
+@article{Liao_2019_Hierarchical,
+ author = {Liao, Qinyi and Berthier, Ludovic},
+ title = {Hierarchical Landscape of Hard Disk Glasses},
+ journal = {Physical Review X},
+ publisher = {American Physical Society (APS)},
+ year = {2019},
+ month = {3},
+ number = {1},
+ volume = {9},
+ pages = {011049},
+ url = {https://doi.org/10.1103%2Fphysrevx.9.011049},
+ doi = {10.1103/physrevx.9.011049}
+}
+
 @article{Maimbourg_2016_Solution,
  author = {Maimbourg, Thibaud and Kurchan, Jorge and Zamponi, Francesco},
  title = {Solution of the Dynamics of Liquids in the Large-Dimensional Limit},
@@ -426,4 +539,32 @@ stochastic localization},
  doi = {10.1103/physrevx.9.011003}
 }
 
+@article{Seguin_2016_Experimental,
+ author = {Seguin, A. and Dauchot, O.},
+ title = {Experimental Evidence of the {Gardner} Phase in a Granular Glass},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2016},
+ month = {11},
+ number = {22},
+ volume = {117},
+ pages = {228001},
+ url = {https://doi.org/10.1103%2Fphysrevlett.117.228001},
+ doi = {10.1103/physrevlett.117.228001}
+}
+
+@article{Xiao_2022_Probing,
+ author = {Xiao, Hongyi and Liu, Andrea J. and Durian, Douglas J.},
+ title = {Probing {Gardner} Physics in an Active Quasithermal Pressure-Controlled Granular System of Noncircular Particles},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2022},
+ month = {6},
+ number = {24},
+ volume = {128},
+ pages = {248001},
+ url = {https://doi.org/10.1103%2Fphysrevlett.128.248001},
+ doi = {10.1103/physrevlett.128.248001}
+}
+
 
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index b1c25ab..23a41d4 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -38,21 +38,29 @@ In this paper we present what we argue is the general replica ansatz for the
 computation of the number of saddles of generic mean-field models,  which we expect to include the Sherrington--Kirkpatrick model. It reproduces  the Parisi result in the limit
 of small temperature for the lowest states, as it should.
 
-To understand the importance of this computation, consider the following situation. When one solves the problem of spheres in large dimensions, one finds that there is
-a transition at a given temperature to a one-step symmetry breaking (1RSB) phase at a Kauzmann temperature,
-and, at a lower temperature,
-another  transition to  a full RSB phase (see \cite{Gross_1985_Mean-field, Gardner_1985_Spin}, the so-called `Gardner' phase \cite{Charbonneau_2014_Fractal}). 
-Now, this transition involves the lowest, equilibrium states. Because they are
-obviously unreachable at any reasonable timescale, an often addressed question
-to ask is: what is the Gardner transition line for higher than equilibrium
-energy-densities? (see, for a review \cite{Berthier_2019_Gardner})  For example,
-when studying `jamming' at zero temperature, the question is posed as to`on
-what side of the 1RSB-FRS transition are the high energy (or low density)
-states reachable dynamically'.   In the present paper we give a concrete strategy to define
-unambiguously such an issue: we consider the local energy minima at a given
-energy and study their number and other properties: the solution involves a
-replica-symmetry breaking scheme that is well-defined, and corresponds directly
-to the topological characteristics of those minima.
+To understand the importance of this computation, consider the following
+situation. When one solves the problem of spheres in large dimensions, one
+finds that there is a transition at a given temperature to a one-step symmetry
+breaking (1RSB) phase at a Kauzmann temperature, and, at a lower temperature,
+another  transition to  a full RSB phase (see \cite{Gross_1985_Mean-field,
+Gardner_1985_Spin}, the so-called `Gardner' phase
+\cite{Charbonneau_2014_Fractal}).  Now, this transition involves the lowest,
+equilibrium states. Because they are obviously unreachable at any reasonable
+timescale, an often addressed question to ask is: what is the Gardner
+transition line for higher than equilibrium energy-densities? This is a
+question whose answers are significant to interpreting the results of myriad
+experiments and simulations \cite{Xiao_2022_Probing, Hicks_2018_Gardner,
+Liao_2019_Hierarchical, Dennis_2020_Jamming, Charbonneau_2015_Numerical,
+Li_2021_Determining, Seguin_2016_Experimental, Geirhos_2018_Johari-Goldstein,
+Hammond_2020_Experimental, Albert_2021_Searching} (see, for a review
+\cite{Berthier_2019_Gardner}).  For example, when studying `jamming' at zero
+temperature, the question is posed as to`on what side of the 1RSB-FRS
+transition are the high energy (or low density) states reachable dynamically'.
+In the present paper we give a concrete strategy to define unambiguously such
+an issue: we consider the local energy minima at a given energy and study their
+number and other properties: the solution involves a replica-symmetry breaking
+scheme that is well-defined, and corresponds directly to the topological
+characteristics of those minima.
 
 
 Perhaps the most interesting application of this computation is in the context  of
@@ -589,7 +597,10 @@ Understanding that $R$ is diagonal, this implies
   \mu^*=\frac1{r_d}+r_df''(1)
 \end{equation}
 which is precisely the condition \eqref{eq:mu.minima}. Therefore, \emph{the
-supersymmetric solution only counts the most common minima} \cite{Annibale_2004_Coexistence}.
+supersymmetric solution counts the most common minima}
+\cite{Annibale_2004_Coexistence}. When minima are not the most common type of
+stationary point, the supersymmetric solution correctly counts minima that
+satisfy \eqref{eq:mu.minima}, but these do not have any special significance.
 
 Inserting the supersymmetric ansatz $D=\hat\beta R$ and $R=r_dI$, one gets
 \begin{equation} \label{eq:diagonal.action}