Displaying similar documents to “On the free energy of a Gaussian disordered mean field model.”

The Gaussian zoo.

Renze, John, Wagon, Stan, Wick, Brian (2001)

Experimental Mathematics

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Geometric influences II: Correlation inequalities and noise sensitivity

Nathan Keller, Elchanan Mossel, Arnab Sen (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand’s lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini–Kalai–Schramm (BKS) noise sensitivity...

Sparse recovery with pre-Gaussian random matrices

Simon Foucart, Ming-Jun Lai (2010)

Studia Mathematica

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For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by ℓ₁-minimization under the optimal condition m ≥ csln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the ℓ₁-norm and the outer norm depends on probability distributions.

Generalized q-deformed Gaussian random variables

Marek Bożejko, Hiroaki Yoshida (2006)

Banach Center Publications

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We produce generalized q-Gaussian random variables which have two parameters of deformation. One of them is, of course, q as for the usual q-deformation. We also investigate the corresponding Wick formulas, which will be described by some joint statistics on pair partitions.

On the proof of the Parisi formula by Guerra and Talagrand

Erwin Bolthausen (2004-2005)

Séminaire Bourbaki

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The Parisi formula is an expression for the limiting free energy of the Sherrington-Kirkpatrick spin glass model, which had first been derived by Parisi using a non-rigorous replica method together with an hierarchical ansatz for the solution of the variational problem. It had become quickly clear that behind the solution, if correct, lies an interesting mathematical structure. The formula has recently been proved by Michel Talagrand based partly on earlier ideas and results by Francesco...

Gaussian Integers

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama (2013)

Formalized Mathematics

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Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic. ...

Robust optimality of Gaussian noise stability

Elchanan Mossel, Joe Neeman (2015)

Journal of the European Mathematical Society

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We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result...

Shift inequalities of Gaussian type and norms of barycentres

F. Barthe, D. Cordero-Erausquin, M. Fradelizi (2001)

Studia Mathematica

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We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition. ...

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Yimin Xiao (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of ( N , d ) -Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the...