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| -rw-r--r-- | bezout.bib | 82 | ||||
| -rw-r--r-- | bezout.tex | 14 |
2 files changed, 91 insertions, 5 deletions
@@ -1,3 +1,30 @@ +@article{Anninos_2016_Disordered, + author = {Anninos, Dionysios and Anous, Tarek and Denef, Frederik}, + title = {Disordered quivers and cold horizons}, + journal = {Journal of High Energy Physics}, + publisher = {Springer Science and Business Media LLC}, + year = {2016}, + month = {12}, + number = {12}, + volume = {2016}, + url = {https://doi.org/10.1007%2Fjhep12%282016%29071}, + doi = {10.1007/jhep12(2016)071} +} + +@article{Antenucci_2015_Complex, + author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.}, + title = {Complex spherical {$2+4$} spin glass: A model for nonlinear optics in random media}, + journal = {Physical Review A}, + publisher = {American Physical Society (APS)}, + year = {2015}, + month = {5}, + number = {5}, + volume = {91}, + pages = {053816}, + url = {https://doi.org/10.1103%2Fphysreva.91.053816}, + doi = {10.1103/physreva.91.053816} +} + @book{Bezout_1779_Theorie, author = {Bézout, Etienne}, title = {Théorie générale des équations algébriques}, @@ -7,6 +34,47 @@ address = {rue S. Jacques, Paris} } +@article{Bray_2007_Statistics, + author = {Bray, Alan J. and Dean, David S.}, + title = {Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {2007}, + month = {4}, + number = {15}, + volume = {98}, + pages = {150201}, + url = {https://doi.org/10.1103%2Fphysrevlett.98.150201}, + doi = {10.1103/physrevlett.98.150201} +} + +@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}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {2004}, + month = {6}, + number = {24}, + volume = {92}, + pages = {240601}, + url = {https://doi.org/10.1103%2Fphysrevlett.92.240601}, + doi = {10.1103/physrevlett.92.240601} +} + +@article{Kac_1943_On, + author = {Kac, M.}, + title = {On the average number of real roots of a random algebraic equation}, + journal = {Bulletin of the American Mathematical Society}, + publisher = {American Mathematical Society}, + year = {1943}, + month = {4}, + number = {4}, + volume = {49}, + pages = {314--320}, + url = {https://projecteuclid.org:443/euclid.bams/1183505112} +} + @article{Nguyen_2014_The, author = {Nguyen, Hoi H. and O'Rourke, Sean}, title = {The Elliptic Law}, @@ -21,6 +89,20 @@ doi = {10.1093/imrn/rnu174} } +@article{Rice_1939_The, + author = {Rice, S. O.}, + title = {The Distribution of the Maxima of a Random Curve}, + journal = {American Journal of Mathematics}, + publisher = {JSTOR}, + year = {1939}, + month = {4}, + number = {2}, + volume = {61}, + pages = {409}, + url = {https://doi.org/10.2307%2F2371510}, + doi = {10.2307/2371510} +} + @article{Weyl_1912_Das, author = {Weyl, Hermann}, title = {Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)}, @@ -46,16 +46,20 @@ The most tractable family of these are the mean-field spherical p-spin models d 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. -This problem has been attacked from several angles: the replica trick to compute the Boltzmann-Gibbs distribution, -a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral) to compute the number of saddle-points of the energy function, and the gradient-descent -- or more generally Langevin -- dynamics staring from a high-energy configuration. -Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit. +This problem has been attacked from several angles: the replica trick to +compute the Boltzmann--Gibbs distribution, 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, and +the gradient-descent -- or more generally Langevin -- dynamics staring from a +high-energy configuration. 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 whose elements are complex normal with $\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$. The motivations for this paper are of two types. On the practical side, there are situations in which complex variables -have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Leuzzi-Crisanti, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos-Anous-Denef} modeling Black Hole horizons in the zero-temperature limit. +have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Antenucci_2015_Complex, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos_2016_Disordered} modeling Black Hole horizons in the zero-temperature limit. There is however a more fundamental reason for this study: we know from experience that extending a problem to the complex plane often uncovers an underlying simplicity that is hidden in the purely real case. Consider, for example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a polynomial of degree $p$ chosen to have simple, known roots. Because we are working in complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. With real @@ -98,7 +102,7 @@ $N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all, of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact. -A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{BrayDean,Fyodorov}, +A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{Bray_2007_Statistics, Fyodorov_2004_Complexity}, so that we may treat the delta-functions and the Hessians as independent. } |