Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Partial results on the convexity of the Parisi functional with PDE approach

Author: Wei-Kuo Chen
Journal: Proc. Amer. Math. Soc. 143 (2015), 3135-3146
MSC (2010): Primary 60K35, 82B44; Secondary 49K20, 39B62
Published electronically: March 13, 2015
MathSciNet review: 3336637
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In a recent work of Panchenko, it was proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies the original proof and presents more general results.

References [Enhancements On Off] (What's this?)

  • [1] Francesco Guerra, Broken replica symmetry bounds in the mean field spin glass model, Comm. Math. Phys. 233 (2003), no. 1, 1-12. MR 1957729 (2003k:82048),
  • [2] Marc Mézard, Giorgio Parisi, and Miguel Angel Virasoro, Spin glass theory and beyond, World Scientific Lecture Notes in Physics, vol. 9, World Scientific Publishing Co. Inc., Teaneck, NJ, 1987. MR 1026102 (91k:82066)
  • [3] Dmitry Panchenko, A question about the Parisi functional, Electron. Comm. Probab. 10 (2005), 155-166 (electronic). MR 2162815 (2006m:82063),
  • [4] Panchenko, D. (2013) The Parisi formula for mixed $ p$-spin models. To appear in Ann. Probab.MR 3189062
  • [5] D. Sherrington and S. Kirkpatrick, (1975) Solvable model of a spin glass. Phys. Rev. Lett. 35 (1975), 1792-1796.
  • [6] Michel Talagrand, The Parisi formula, Ann. of Math. (2) 163 (2006), no. 1, 221-263. MR 2195134 (2007m:82041),
  • [7] Michel Talagrand, Parisi measures, J. Funct. Anal. 231 (2006), no. 2, 269-286. MR 2195333 (2007h:82035),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60K35, 82B44, 49K20, 39B62

Retrieve articles in all journals with MSC (2010): 60K35, 82B44, 49K20, 39B62

Additional Information

Wei-Kuo Chen
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Keywords: Parisi formula, Sherrington-Kirkpatrick model
Received by editor(s): August 30, 2013
Received by editor(s) in revised form: October 25, 2013
Published electronically: March 13, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society