On one property of Derrida–Ruelle cascades

Dmitry Panchenko; Michel Talagrand

doi:10.1016/j.crma.2007.10.035

Probability Theory

On one property of Derrida–Ruelle cascades
[Sur une propriété des cascades de Derrida–Ruelle]

Présenté par : Gilles Pisier

Dmitry Panchenko ¹ ; Michel Talagrand ²

¹ Texas A&M University, College Station, TX 77843, USA
² Équipe d'analyse de l'institut mathématique de Jussieu, 4, place Jussieu, 75230 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 653-656.

Résumé
Abstract

Nous présentons une nouvelle propriété des processus ponctuels de Poisson–Dirichlet et des cascades de Derrida–Ruelle. Cela nous permet d'exprimer l'interpolation de Guerra dans le modèle de verres de spin de Sherrington–Kirkpatric entièrement dans le langage des cascades.

We present new results about Poisson–Dirichlet point processes and Derrida–Ruelle cascades that are motivated by applications to mean-field spin glass models and, in particular, by the attempt to express Guerra's interpolation in the Sherrington–Kirkpatrick model entirely in the language of the cascades.

Reçu le : 2007-09-25
Accepté le : 2007-10-16
Publié le : 2007-11-26

DOI : 10.1016/j.crma.2007.10.035

Affiliations des auteurs :

Dmitry Panchenko ¹ ; Michel Talagrand ²

¹ Texas A&M University, College Station, TX 77843, USA
² Équipe d'analyse de l'institut mathématique de Jussieu, 4, place Jussieu, 75230 Paris cedex 05, France

@article{CRMATH_2007__345_11_653_0,
     author = {Dmitry Panchenko and Michel Talagrand},
     title = {On one property of {Derrida{\textendash}Ruelle} cascades},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {653--656},
     publisher = {Elsevier},
     volume = {345},
     number = {11},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.035},
     language = {en},
}

TY  - JOUR
AU  - Dmitry Panchenko
AU  - Michel Talagrand
TI  - On one property of Derrida–Ruelle cascades
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 653
EP  - 656
VL  - 345
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2007.10.035
LA  - en
ID  - CRMATH_2007__345_11_653_0
ER  -

%0 Journal Article
%A Dmitry Panchenko
%A Michel Talagrand
%T On one property of Derrida–Ruelle cascades
%J Comptes Rendus. Mathématique
%D 2007
%P 653-656
%V 345
%N 11
%I Elsevier
%R 10.1016/j.crma.2007.10.035
%G en
%F CRMATH_2007__345_11_653_0

Dmitry Panchenko; Michel Talagrand. On one property of Derrida–Ruelle cascades. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 653-656. doi : 10.1016/j.crma.2007.10.035. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.035/

Bibliographie
Cité par

[1] L.-P. Arguin Spin glass computations and Ruelle's probability cascades, J. Stat. Phys., Volume 126 (2007) no. 4–5, pp. 951-976

[2] M. Aizenman; R. Sims; S. Starr An extended variational principle for the SK spin-glass model, Phys. Rev. B, Volume 68 (2003), p. 214403

[3] E. Bolthausen; A.-S. Sznitman On Ruelle's probability cascades and an abstract cavity method, Comm. Math. Phys., Volume 197 (1998) no. 2, pp. 247-276

[4] A. Bovier; I. Kurkova Derrida's generalized random energy models I, Models with finitely many hierarchies, Ann. Inst. H. Poincaré Probab. Statist., Volume 40 (2004), pp. 439-480

[5] B. Derrida Random-energy model: An exactly solvable model of disordered systems, Phys. Rev. B, Volume 24 (1981), pp. 2613-2626

[6] B. Derrida A generalization of the random energy model which includes correlations between energies, J. Phys. Lett., Volume 46 (1985), p. L401-L-407

[7] F. Guerra Broken replica symmetry bounds in the mean field spin glass model, Commun. Math. Phys., Volume 233 (2003) no. 1, pp. 1-12

[8] J.F.C. Kingman Poisson Processes, Oxford University Press, New York, 1993

[9] D. Panchenko; M. Talagrand Bounds for diluted mean-fields spin glass models, Probab. Theory Related Fields, Volume 130 (2004) no. 3, pp. 319-336

[10] D. Panchenko, M. Talagrand, Guerra's interpolation using Derrida–Ruelle cascades, preprint, arxiv:0709.1514

[11] D. Ruelle A mathematical reformulation of Derrida's REM and GREM, Commun. Math. Phys., Volume 108 (1987) no. 2, pp. 225-239

[12] A. Ruzmaikina; M. Aizenman Characterization of invariant measures at the leading edge for competing particle systems, Ann. Probab., Volume 33 (2005) no. 1, pp. 82-113

[13] M. Talagrand Spin Glasses: A Challenge for Mathematicians, Springer-Verlag, 2003

[14] M. Talagrand Parisi formula, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 221-263

Benjamin Bonnefont; Michel Pain; Olivier Zindy Branching Brownian Motion Versus Random Energy Model in the Supercritical Phase: Overlap Distribution and Temperature Susceptibility, Journal of Statistical Physics, Volume 192 (2025) no. 2 | DOI:10.1007/s10955-025-03394-0
Jean-Christophe Mourrat The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space, Canadian Journal of Mathematics, Volume 74 (2022) no. 3, p. 607 | DOI:10.4153/s0008414x21000031
Benjamin Bonnefont The overlap distribution at two temperatures for the branching Brownian motion, Electronic Journal of Probability, Volume 27 (2022) no. none | DOI:10.1214/22-ejp841
Michel Pain; Olivier Zindy Two-temperatures overlap distribution for the 2D discrete Gaussian free field, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 57 (2021) no. 2 | DOI:10.1214/20-aihp1091
Adriano Barra; Pierluigi Contucci; Emanuele Mingione; Daniele Tantari Multi-Species Mean Field Spin Glasses. Rigorous Results, Annales Henri Poincaré, Volume 16 (2015) no. 3, p. 691 | DOI:10.1007/s00023-014-0341-5
Bartlomiej Blaszczyszyn; Mohamed Kadhem Karray; H. Paul Keeler Wireless Networks Appear Poissonian Due to Strong Shadowing, IEEE Transactions on Wireless Communications, Volume 14 (2015) no. 8, p. 4379 | DOI:10.1109/twc.2015.2420099
Dmitry Panchenko Spin glass models from the point of view of spin distributions, The Annals of Probability, Volume 41 (2013) no. 3A | DOI:10.1214/11-aop696
Charles Bordenave; Djalil Chafaï Around the circular law, Probability Surveys, Volume 9 (2012) no. none | DOI:10.1214/11-ps183
Charles Bordenave; Pietro Caputo; Djalil Chafaï Spectrum of Non-Hermitian Heavy Tailed Random Matrices, Communications in Mathematical Physics, Volume 307 (2011) no. 2, p. 513 | DOI:10.1007/s00220-011-1331-9
Dmitry Panchenko A connection between the Ghirlanda–Guerra identities and ultrametricity, The Annals of Probability, Volume 38 (2010) no. 1 | DOI:10.1214/09-aop484
Anton Bovier; Anton Klimovsky The Aizenman-Sims-Starr and Guerras schemes for the SK model with multidimensional spins, Electronic Journal of Probability, Volume 14 (2009) no. none | DOI:10.1214/ejp.v14-611

Cité par 11 documents. Sources : Crossref

Commentaires - Politique