# On the proof of the Parisi formula by Guerra and Talagrand

Séminaire Bourbaki (2004-2005)

- Volume: 47, page 349-378
- ISSN: 0303-1179

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topBolthausen, Erwin. "On the proof of the Parisi formula by Guerra and Talagrand." Séminaire Bourbaki 47 (2004-2005): 349-378. <http://eudml.org/doc/252149>.

@article{Bolthausen2004-2005,

abstract = {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 Guerra. The talk will try to explain why the problem is mathematically interesting, and sketch the ideas of Guerra and Talagrand. It should be emphasized that despite the fact that the formula is proved, many things remain still quite mysterious.},

author = {Bolthausen, Erwin},

journal = {Séminaire Bourbaki},

keywords = {verre de spin; modèle de Sherrington-Kirkpatrick; énergie libre; brisure de la symétrie des répliques},

language = {eng},

pages = {349-378},

publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},

title = {On the proof of the Parisi formula by Guerra and Talagrand},

url = {http://eudml.org/doc/252149},

volume = {47},

year = {2004-2005},

}

TY - JOUR

AU - Bolthausen, Erwin

TI - On the proof of the Parisi formula by Guerra and Talagrand

JO - Séminaire Bourbaki

PY - 2004-2005

PB - Association des amis de Nicolas Bourbaki, Société mathématique de France

VL - 47

SP - 349

EP - 378

AB - 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 Guerra. The talk will try to explain why the problem is mathematically interesting, and sketch the ideas of Guerra and Talagrand. It should be emphasized that despite the fact that the formula is proved, many things remain still quite mysterious.

LA - eng

KW - verre de spin; modèle de Sherrington-Kirkpatrick; énergie libre; brisure de la symétrie des répliques

UR - http://eudml.org/doc/252149

ER -

## References

top- [1] M. Aizenman, J. Lebowitz & D. Ruelle – “Some rigorous results on the Sherrington-Kirkpatrick model”, Comm. Math. Phys.112 (1987), p. 3–20. Zbl1108.82312MR904135
- [2] M. Aizenman, R. Sims & S.L. Starr – “An extended variational principle for the SK spin-glass model”, Phys. Rev. B68 (2003), p. 214–403.
- [3] J.R.L. de Almeida & D.J. Thouless – “Stability of the Sherrington-Kirkpatrick solution of spin glasses”, J. Phys. A 11 (1978), p. 983.
- [4] E. Bolthausen & N. Kistler – “On a non-hierarchical version of the Generalized Random Energy Model”, Ann. Appl. Prob.16 (2006), p. 1–14. Zbl1100.60026MR2209333
- [5] E. Bolthausen & A.-S. Sznitman – “On Ruelle’s probability cascades and an abstract cavity method”, Comm. Math. Phys.197 (1998), p. 247–276. Zbl0927.60071MR1652734
- [6] A. Bovier & I. Kurkova – “Derrida’s Generalized Random Energy Models I & II”, Annales de l’Institut Henri Poincaré40 (2004), p. 439–495. Zbl1121.82020MR2070334
- [7] B. Derrida – “Random energy model: An exactly solvable model of disordered systems”, Phys. Rev. B24 (1981), p. 2613–2626. Zbl1323.60134MR627810
- [8] —, “A generalization of the random energy model that includes correlations between the energies”, J. Physique. Lett46 (1986), p. 401–407.
- [9] F. Guerra – “Replica broken bounds in the mean field spin glass model”, Comm. Math. Phys.233 (2003), p. 1–12. Zbl1013.82023MR1957729
- [10] F. Guerra & F. L. Toninelli – “Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model”, J. Math. Phys.43 (2002), p. 3704–3716. Zbl1060.82023MR1908695
- [11] —, “The thermodynamic limit in mean field spin glass models”, Comm. Math. Phys.230 (2002), p. 71–79. Zbl1004.82004MR1930572
- [12] M. Ledoux – “The Concentration of Measure Phenomenon”, vol. 89, AMS, 2002. Zbl0995.60002MR1849347
- [13] M. Mézard, G. Parisi & M.A. Virasoro – Spin Glass Theory and Beyond, World Scientific, 1987. Zbl0992.82500MR1026102
- [14] H. Nishimori – Statistical Physics of Spin Glasses and Information Processing, Oxford Science Publications, 1999. Zbl1103.82002
- [15] G. Parisi – “A sequence of approximate solutions to the S-K model for spin glasses”, J. Phys. A 13 L-115 (1980).
- [16] D. Ruelle – “A mathematical reformulation of Derrida’s REM and GREM”, Comm. Math. Phys.108 (1987), p. 225–239. Zbl0617.60100MR875300
- [17] A. Ruzmaikina & M. Aizenman – “Characterization of invariant measures at the leading edge for competing particle systems”, Ann. Prob.33 (2005), p. 83–113. Zbl1096.60042MR2118860
- [18] D. Sherrington & S. Kirkpatrick – “Solvable model of a spin glass”, Phys. Rev. Lett.35 (1972), p. 1792–1796.
- [19] M. Talagrand – Spin Glasses: A Challenge for Mathematicians, Springer, Heidelberg, 2003. Zbl1033.82002MR1993891
- [20] —, “The Parisi formula”, Ann. Math.163 (2006), p. 221–263. Zbl1137.82010MR2195134

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