The geometry of the Gibbs measure of pure spherical spin glasses
TLDR
In this paper, the Gibbs measure splits into infinitesimal spherical ''bands' centered at deep minima, playing the role of so-called pure states, and the transition rate to disorder chaos is analyzed.Abstract:
We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature.read more
Citations
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The Landscape of the Spiked Tensor Model
TL;DR: In this paper, the authors consider the problem of estimating a rank-one tensor in Gaussian noise and show that the expected number of critical points and local maxima of this objective function is exponential in the dimensions of the unit sphere and give exact formulas for the exponential growth rate.
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The extremal process of critical points of the pure p -spin spherical spin glass model
Eliran Subag,Ofer Zeitouni +1 more
TL;DR: In this paper, the extremal point process of critical points is studied, i.e., the point process associated to all critical values in the vicinity of the ground-state.
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Following the ground-states of full-RSB spherical spin glasses
TL;DR: For spherical spin glasses whose Parisi distribution has support of the form [0, q], this work construct paths from the origin to the sphere that consistently remain close to the ground‐state energy on the sphere of corresponding radius using a greedy strategy.
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Free energy landscapes in spherical spin glasses
TL;DR: In this paper, the authors introduce and analyze free energy landscapes defined by associating to any point inside the sphere a free energy calculated on a thin spherical band around it, using many orthogonal replicas.
References
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Book
Large Deviations Techniques and Applications
Amir Dembo,Ofer Zeitouni +1 more
TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.
Book
Spectral Analysis of Large Dimensional Random Matrices
Zhidong Bai,Jack W. Silverstein +1 more
TL;DR: Wigner Matrices and Semicircular Law for Hadamard products have been used in this article for spectral separations and convergence rates of ESD for linear spectral statistics.
Book
Random Fields and Geometry
Robert J. Adler,Jonathan Taylor +1 more
TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.
Journal ArticleDOI
Solution of 'Solvable model of a spin glass'
TL;DR: In this paper, the Sherrmgton-Kirkpatrick model of a spin glass is solved by a mean field technique which is probably exact in the limit of infinite range interactions.
Journal ArticleDOI
Order parameter for spin-glasses
TL;DR: In this paper, an order parameter for spin-glasses is defined in a clear physical way: it is a function on the interval 0-1 and it is related to the probability distribution of the overlap of the magnetization in different states of the system.