From e89513ed62967929dcf09b2944c8301451366e99 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 9 Dec 2020 14:22:49 +0100 Subject: Updated bibliography with DOIs and fixed label. --- bezout.bib | 182 +++++++++++++++++++++++++++++++++---------------------------- bezout.tex | 18 +++--- 2 files changed, 108 insertions(+), 92 deletions(-) diff --git a/bezout.bib b/bezout.bib index a23955b..5b61d1e 100644 --- a/bezout.bib +++ b/bezout.bib @@ -26,6 +26,34 @@ doi = {10.1103/physreva.91.053816} } +@article{Auffinger_2012_Random, + author = {Auffinger, Antonio and Arous, Gérard Ben and Černý, Jiří}, + title = {Random Matrices and Complexity of Spin Glasses}, + journal = {Communications on Pure and Applied Mathematics}, + publisher = {Wiley}, + year = {2012}, + month = {9}, + number = {2}, + volume = {66}, + pages = {165--201}, + url = {https://doi.org/10.1002%2Fcpa.21422}, + doi = {10.1002/cpa.21422} +} + +@article{Auffinger_2013_Complexity, + author = {Auffinger, Antonio and Arous, Gerard Ben}, + title = {Complexity of random smooth functions on the high-dimensional sphere}, + journal = {The Annals of Probability}, + publisher = {Institute of Mathematical Statistics}, + year = {2013}, + month = {11}, + number = {6}, + volume = {41}, + pages = {4214--4247}, + url = {https://doi.org/10.1214%2F13-aop862}, + doi = {10.1214/13-aop862} +} + @book{Bezout_1779_Theorie, author = {Bézout, Etienne}, title = {Théorie générale des équations algébriques}, @@ -63,6 +91,76 @@ doi = {10.1103/physrevlett.98.150201} } +@article{Cartwright_2013_The, + author = {Cartwright, Dustin and Sturmfels, Bernd}, + title = {The number of eigenvalues of a tensor}, + journal = {Linear Algebra and its Applications}, + publisher = {Elsevier BV}, + year = {2013}, + month = {1}, + number = {2}, + volume = {438}, + pages = {942--952}, + url = {https://doi.org/10.1016%2Fj.laa.2011.05.040}, + doi = {10.1016/j.laa.2011.05.040} +} + +@article{Castellani_2005_Spin-glass, + author = {Castellani, Tommaso and Cavagna, Andrea}, + title = {Spin-glass theory for pedestrians}, + journal = {Journal of Statistical Mechanics: Theory and Experiment}, + publisher = {IOP Publishing}, + year = {2005}, + month = {5}, + number = {05}, + volume = {2005}, + pages = {P05012}, + url = {https://doi.org/10.1088%2F1742-5468%2F2005%2F05%2Fp05012}, + doi = {10.1088/1742-5468/2005/05/p05012} +} + +@article{Crisanti_1992_The, + author = {Crisanti, A. and Sommers, H. -J.}, + title = {The spherical $p$-spin interaction spin glass model: the statics}, + journal = {Zeitschrift für Physik B Condensed Matter}, + publisher = {Springer Science and Business Media LLC}, + year = {1992}, + month = {10}, + number = {3}, + volume = {87}, + pages = {341--354}, + url = {https://doi.org/10.1007%2Fbf01309287}, + doi = {10.1007/bf01309287} +} + +@article{Crisanti_1995_Thouless-Anderson-Palmer, + author = {Crisanti, A. and Sommers, H. -J.}, + title = {Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model}, + journal = {Journal de Physique I}, + publisher = {EDP Sciences}, + year = {1995}, + month = {7}, + number = {7}, + volume = {5}, + pages = {805--813}, + url = {https://doi.org/10.1051%2Fjp1%3A1995164}, + doi = {10.1051/jp1:1995164} +} + +@article{Cugliandolo_1993_Analytical, + author = {Cugliandolo, L. F. and Kurchan, J.}, + title = {Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {1993}, + month = {7}, + number = {1}, + volume = {71}, + pages = {173--176}, + url = {https://doi.org/10.1103%2Fphysrevlett.71.173}, + doi = {10.1103/physrevlett.71.173} +} + @article{Fyodorov_2004_Complexity, author = {Fyodorov, Yan V.}, title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices}, @@ -132,86 +230,4 @@ doi = {10.1007/bf01456804} } -@article{auffinger2013random, - title={Random matrices and complexity of spin glasses}, - author={Auffinger, Antonio and Arous, G{\'e}rard Ben and {\v{C}}ern{\`y}, Ji{\v{r}}{\'\i}}, - journal={Communications on Pure and Applied Mathematics}, - volume={66}, - number={2}, - pages={165--201}, - year={2013}, - publisher={Wiley Online Library} -} - -@article{cartwright2013number, - title={The number of eigenvalues of a tensor}, - author={Cartwright, Dustin and Sturmfels, Bernd}, - journal={Linear algebra and its applications}, - volume={438}, - number={2}, - pages={942--952}, - year={2013}, - publisher={Elsevier} -} - -@article{auffinger2013complexity, - title={Complexity of random smooth functions on the high-dimensional sphere}, - author={Auffinger, Antonio and Arous, Gerard Ben and others}, - journal={The Annals of Probability}, - volume={41}, - number={6}, - pages={4214--4247}, - year={2013}, - publisher={Institute of Mathematical Statistics} -} -@article{fyodorov2004complexity, - title={Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices}, - author={Fyodorov, Yan V}, - journal={Physical review letters}, - volume={92}, - number={24}, - pages={240601}, - year={2004}, - publisher={APS} -} -@article{crisanti1995thouless, - title={Thouless-Anderson-Palmer approach to the spherical p-spin spin glass model}, - author={Crisanti, Andrea and Sommers, H-J}, - journal={Journal de Physique I}, - volume={5}, - number={7}, - pages={805--813}, - year={1995}, - publisher={EDP Sciences} -} -@article{crisanti1992sphericalp, - title={The sphericalp-spin interaction spin glass model: the statics}, - author={Crisanti, Andrea and Sommers, H-J}, - journal={Zeitschrift f{\"u}r Physik B Condensed Matter}, - volume={87}, - number={3}, - pages={341--354}, - year={1992}, - publisher={Springer} -} -@article{castellani2005spin, - title={Spin-glass theory for pedestrians}, - author={Castellani, Tommaso and Cavagna, Andrea}, - journal={Journal of Statistical Mechanics: Theory and Experiment}, - volume={2005}, - number={05}, - pages={P05012}, - year={2005}, - publisher={IOP Publishing} -} - -@article{cugliandolo1993analytical, - title={Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model}, - author={Cugliandolo, Leticia F and Kurchan, Jorge}, - journal={Physical Review Letters}, - volume={71}, - number={1}, - pages={173}, - year={1993}, - publisher={APS} -} + diff --git a/bezout.tex b/bezout.tex index 5fc340a..1a12025 100644 --- a/bezout.tex +++ b/bezout.tex @@ -39,22 +39,22 @@ different topological properties. Spin-glasses have long been considered the paradigm of `complex landscapes' of many variables, a subject that includes Neural Networks and optimization problems, most notably Constraint Satisfaction ones. -The most tractable family of these are the mean-field spherical p-spin models \cite{crisanti1992sphericalp} (for a review see \cite{castellani2005spin}) +The most tractable family of these are the mean-field spherical p-spin models \cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass}) defined by the energy: \begin{equation} \label{eq:bare.hamiltonian} H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, \end{equation} where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall study here. -Also in the Algebra \cite{cartwright2013number} and Probability literature \cite{auffinger2013complexity,auffinger2013random}. +Also in the Algebra \cite{Cartwright_2013_The} and Probability literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. This problem has been attacked from several angles: the replica trick to -compute the Boltzmann--Gibbs distribution\cite{crisanti1992sphericalp}, a Kac--Rice \cite{Kac_1943_On, +compute the Boltzmann--Gibbs distribution\cite{Crisanti_1992_The}, a Kac--Rice \cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the Fadeev--Popov integral) to compute the number of saddle-points of the energy function -\cite{crisanti1995thouless}, and +\cite{Crisanti_1995_Thouless-Anderson-Palmer}, and the gradient-descent -- or more generally Langevin -- dynamics staring from a -high-energy configuration \cite{cugliandolo1993analytical}. Thanks to the relative simplicity of the energy, +high-energy configuration \cite{Cugliandolo_1993_Analytical}. Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit. In this paper we shall extend the study to the case where $z\in\mathbb C^N$ are and $J$ is a symmetric tensor @@ -325,13 +325,14 @@ Consider for example the ground-state energy for given $a$, that is, the energy The complexity of the pure 3-spin model at $\epsilon=0$ as a function of $a$ at several values of $\kappa$. The dashed line shows $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. - } + } \label{fig:complexity} \end{figure} -{\color{teal} {\bf somewhere} In Figure \ref{desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$. +\textcolor{teal}{ {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$. For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation is more interesting. In the range of values of $\Re H_0$ where there are real solutions there are solutions all the way down to $a=1$: this is only possible if the density of solutions diverges at this value: this is natural, since. +} \begin{figure}[htpb]\label{desert} @@ -341,10 +342,9 @@ all the way down to $a=1$: this is only possible if the density of solutions div The minimum value of $a$ for which the complexity is positive as a function of (real) energy $\epsilon$ for the pure 3-spin model at several values of $\kappa$. - } + } \label{fig:desert} \end{figure} -} \bibliographystyle{apsrev4-2} \bibliography{bezout} -- cgit v1.2.3-70-g09d2