Comptes Rendus
Probability Theory
On one property of Derrida–Ruelle cascades
Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 653-656.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.10.035
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/

[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

Cited by Sources:

Comments - Policy


Articles of potential interest

On Guerra's broken replica-symmetry bound

Michel Talagrand

C. R. Math (2003)


The Ghirlanda–Guerra identities for mixed p-spin model

Dmitry Panchenko

C. R. Math (2010)


A deletion-invariance property for random measures satisfying the Ghirlanda–Guerra identities

Dmitry Panchenko

C. R. Math (2011)