diff options
| -rw-r--r-- | .gitignore | 1 | ||||
| -rw-r--r-- | frsb_kac-rice.bib | 141 | ||||
| -rw-r--r-- | frsb_kac-rice.tex | 43 |
3 files changed, 169 insertions, 16 deletions
@@ -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} |