The Parisi ultrametricity conjecture

In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $p$-spin models, for which Gibbs' measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.

[1]  M. Aizenman,et al.  On the structure of quasi-stationary competing particle systems , 2007, 0709.2901.

[2]  G. Parisi.  A sequence of approximated solutions to the S-K model for spin glasses , 1980 .

[3]  Michel Talagrand,et al.  Construction of pure states in mean field models for spin glasses , 2010 .

[4]  M. Talagrand.  The parisi formula , 2006 .

[5]  M. Aizenman,et al.  On the Stability of the Quenched State in Mean-Field Spin-Glass Models , 1997, cond-mat/9712129.

[6]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[7]  Giorgio Parisi,et al.  Infinite Number of Order Parameters for Spin-Glasses , 1979 .

[8]  M. Talagrand.  Mean Field Models for Spin Glasses , 2011 .

[9]  D. Panchenko.  The Ghirlanda-Guerra identities for mixed p-spin model , 2010, 1002.2190.

[10]  V. Sudakov,et al.  Gram-de finetti matrices , 1984 .

[11]  Giorgio Parisi,et al.  The Nature of the Spin Glass Phase , 1987 .

[12]  M. Mézard,et al.  Replica symmetry breaking and the nature of the spin glass phase , 1984 .

[13]  M. Talagrand,et al.  Spin Glasses: A Challenge for Mathematicians , 2003 .

[14]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[15]  F. Guerra,et al.  General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity , 1998, cond-mat/9807333.

[16]  F. Guerra.  Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model , 2002, cond-mat/0205123.

[17]  Tim Austin.  Mean field models for spin glasses , 2012 .

[18]  M. Talagrand.  Advanced replica-symmetry and low temperature , 2011 .

[19]  D. Panchenko.  Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case , 2011, 1106.3984.

[20]  M. Mézard,et al.  Nature of the Spin-Glass Phase , 1984 .

[21]  D. Panchenko.  A Unified Stability Property in Spin Glasses , 2011, 1106.3954.

[22]  Dmitry Panchenko,et al.  A connection between the Ghirlanda--Guerra identities and ultrametricity , 2008, 0810.0743.