TY - JOUR

T1 - Spectral Gap Estimates in Mean Field Spin Glasses

AU - Ben Arous, Gérard

AU - Jagannath, Aukosh

N1 - Funding Information:
Acknowledgements. The authors would like to thank G. Biroli, C. Cammarota, and R. Gheissari for many helpful discussions. The authors would also like to thank D. Panchenko and an anonymous referee for drawing their attention to an issue in an earlier version of this paper, where we erroneously extended these arguments to non-convex ξ. It remains a very interesting question to extend these results to this regime. The authors would like to thank A. Montanari and G. Semerjian for drawing their attention to [71]. This research was conducted while G.BA. was supported by NSF DMS1209165, BSF 2014019 and A.J. was supported by NSF OISE-1604232.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for reversible nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally, in the case of Ising spins, Panchenko’s recent rigorous calculation (Panchenko in Ann Probab 46(2):865–896, 2018) of the free energy for a system of “two real replica” enables us to prove a quenched LDP for the overlap distribution, which gives us a wider criterion for slow mixing directly related to the Franz–Parisi–Virasoro approach (Franz et al. in J Phys I 2(10):1869–1880, 1992; Kurchan et al. J Phys I 3(8):1819–1838, 1993). This condition holds in a wider range of temperatures.

AB - We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for reversible nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally, in the case of Ising spins, Panchenko’s recent rigorous calculation (Panchenko in Ann Probab 46(2):865–896, 2018) of the free energy for a system of “two real replica” enables us to prove a quenched LDP for the overlap distribution, which gives us a wider criterion for slow mixing directly related to the Franz–Parisi–Virasoro approach (Franz et al. in J Phys I 2(10):1869–1880, 1992; Kurchan et al. J Phys I 3(8):1819–1838, 1993). This condition holds in a wider range of temperatures.

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U2 - 10.1007/s00220-018-3152-6

DO - 10.1007/s00220-018-3152-6

M3 - Article

AN - SCOPUS:85047142270

SN - 0010-3616

VL - 361

SP - 1

EP - 52

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 1

ER -