## The Annals of Probability

### Thouless–Anderson–Palmer equations for generic $p$-spin glasses

#### Abstract

We study the Thouless–Anderson–Palmer (TAP) equations for spin glasses on the hypercube. First, using a random, approximately ultrametric decomposition of the hypercube, we decompose the Gibbs measure, $\langle \cdot \rangle_{N}$, into a mixture of conditional laws, $\langle \cdot \rangle_{\alpha,N}$. We show that the TAP equations hold for the spin at any site with respect to $\langle \cdot \rangle_{\alpha,N}$ simultaneously for all $\alpha$. This result holds for generic models provided that the Parisi measure of the model has a jump at the top of its support.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2230-2256.

Dates
Revised: August 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205708

Digital Object Identifier
doi:10.1214/18-AOP1307

Mathematical Reviews number (MathSciNet)
MR3980920

#### Citation

Auffinger, Antonio; Jagannath, Aukosh. Thouless–Anderson–Palmer equations for generic $p$-spin glasses. Ann. Probab. 47 (2019), no. 4, 2230--2256. doi:10.1214/18-AOP1307. https://projecteuclid.org/euclid.aop/1562205708

#### References

• [1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York.
• [2] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080–1113.
• [3] Auffinger, A. and Chen, W.-K. (2015). On properties of Parisi measures. Probab. Theory Related Fields 161 817–850.
• [4] Auffinger, A. and Chen, W.-K. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
• [5] Auffinger, A. and Jagannath, A. (2019). On spin distributions for generic $p$-spin models. J. Stat. Phys. 174 316–332.
• [6] Bolthausen, E. (2014). An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Comm. Math. Phys. 325 333–366.
• [7] Chatterjee, S. (2010). Spin glasses and Stein’s method. Probab. Theory Related Fields 148 567–600.
• [8] Jagannath, A. (2017). Approximate ultrametricity for random measures and applications to spin glasses. Comm. Pure Appl. Math. 70 611–664.
• [9] Panchenko, D. (2010). The Ghirlanda–Guerra identities for mixed $p$-spin model. C. R. Math. Acad. Sci. Paris 348 189–192.
• [10] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
• [11] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer Monographs in Mathematics. Springer, New York.
• [12] Panchenko, D. (2015). Hierarchical exchangeability of pure states in mean field spin glass models. Probab. Theory Related Fields 161 619–650.
• [13] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models. Ergebnisse der Mathematik und Ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 46. Springer, Berlin.
• [14] Talagrand, M. (2010). Construction of pure states in mean field models for spin glasses. Probab. Theory Related Fields 148 601–643.
• [15] Thouless, D. J., Anderson, P. W. and Palmer, R. G. (1977). Solution of ‘solvable model of a spin glass’. Philos. Mag. 35 593–601.