Journal of Statistical Physics

, Volume 165, Issue 2, pp 185–224 | Cite as

Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model

  • Jinho Baik
  • Ji Oon Lee


We consider the fluctuations of the free energy for the 2-spin spherical Sherrington–Kirkpatrick model with no magnetic field. We show that the law of the fluctuations converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy–Widom distribution when the temperature is below the critical temperature. The orders of the fluctuations are markedly different in these two regimes. A universality of the limit law is also proved.


Free energy Spherical SK model Tracy–Widom distribution 



We would like to thank Tuca Auffinger, Zhidong Bai, Paul Bourgade, Joe Conlon, Dmitry Panchenko, and Jian-feng Yao for several useful communications. Ji Oon Lee is grateful to the Department of Mathematics, University of Michigan, Ann Arbor, for their kind hospitality during the academic year 2014–2015. The work of Jinho Baik was supported in part by NSF Grants DMS1361782. The work of Ji Oon Lee was supported in part by Samsung Science and Technology Foundation Project No. SSTF-BA1402-04.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Mathematical SciencesKAISTDaejeonKorea

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