On multiple peaks and moderate deviations for the supremum of a Gaussian field.

*(English)*Zbl 1344.60039Authors’ abstract: We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns the phenomenon of multiple peaks. Consider a centered Gaussian field whose sites have variance at most 1, and let \(\rho\) be the standard deviation of its supremum. A theorem of S. Chatterjee [“Chaos, concentration and multiple valleys”, Preprint, arXiv:0810.4221] states that when such a Gaussian field is superconcentrated (i.e., \(\rho\ll 1\)), it typically attains values near its maximum on multiple almost-orthogonal sites and is said to exhibit multiple peaks. We improve his theorem in two respects: (i) the number of peaks attained by our bound is of the order \(\exp(c/\rho^{2})\) (as opposed to Chatterjee’s polynomial bound in \(1/\rho\)) and (ii) our bound does not assume that the correlations are nonnegative. We also prove a similar result based on superconcentration of the free energy. As primary applications, we infer that for the S-K spin glass model on the \(n\)-hypercube and directed polymers on \(\mathbb{Z}_{n}^{2}\), there are polynomially (in \(n\)) many almost-orthogonal sites that achieve values near their respective maxima. {

} Our second theorem gives an upper bound on moderate deviations for the supremum of a general Gaussian field. While the Gaussian isoperimetric inequality implies a sub-Gaussian concentration bound for the supremum, we show that the exponent in that bound can be improved under the assumption that the expectation of the supremum is of the same order as that of the independent case.

} Our second theorem gives an upper bound on moderate deviations for the supremum of a general Gaussian field. While the Gaussian isoperimetric inequality implies a sub-Gaussian concentration bound for the supremum, we show that the exponent in that bound can be improved under the assumption that the expectation of the supremum is of the same order as that of the independent case.

Reviewer: Nikolai N. Leonenko (Cardiff)

##### MSC:

60G15 | Gaussian processes |

60G60 | Random fields |

60G70 | Extreme value theory; extremal stochastic processes |

60F10 | Large deviations |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |

##### Keywords:

Gaussian fields; extreme values; moderate deviations; multiple peaks; isoperimetric inequality; spin glass model; statistical mechanics type models; sub-Gaussian bound
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\textit{J. Ding} et al., Ann. Probab. 43, No. 6, 3468--3493 (2015; Zbl 1344.60039)

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##### References:

[1] | Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 207-216. · Zbl 0292.60004 |

[2] | Castellana, M. and Zarinelli, E. (2011). Role of tracy-widom distribution in finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass. Phys. Rev. B 84 144417. |

[3] | Chatterjee, S. (2008). Chaos, concentration, and multiple valleys. Preprint. Available at . arXiv:0810.4221 |

[4] | Chatterjee, S. (2009). Disorder chaos and multiple valleys in spin glasses. Preprint. Available at . arXiv:0907.3381v4 |

[5] | Chatterjee, S., Dembo, A. and Ding, J. (2013). On level sets of Gaussian fields. Preprint. Available at . arXiv:1310.5175 |

[6] | Cianchi, A., Fusco, N., Maggi, F. and Pratelli, A. (2011). On the isoperimetric deficit in Gauss space. Amer. J. Math. 133 131-186. · Zbl 1219.28005 |

[7] | Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330. · Zbl 0188.20502 |

[8] | Eldan, R. (2013). A two-sided estimate for the Gaussian noise stability deficit. Preprint. Available at . · Zbl 1323.60035 |

[9] | Fernique, X. (1971). Régularité de processus gaussiens. Invent. Math. 12 304-320. · Zbl 0217.21104 |

[10] | Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889-892. · Zbl 1101.82329 |

[11] | Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 |

[12] | Madaule, T. (2013). Maximum of a log-correlated Gaussian field. Preprint. Available at . arXiv:1307.1365 · Zbl 1329.60138 |

[13] | Mossel, E. and Neeman, J. (2015). Robust dimension free isoperimetry in Gaussian space. Ann. Probab. 43 971-991. · Zbl 1320.60063 |

[14] | Mossel, E. and Neeman, J. (2015). Robust optimality of Gaussian noise stability. J. Eur. Math. Soc. ( JEMS ) 17 433-482. · Zbl 1384.60062 |

[15] | Palassini, M. (2008). Ground-state energy fluctuations in the Sherrington-Kirkpatrick model. J. Stat. Mech. Theory Exp. 2008 P10005. |

[16] | Parisi, G. (1980). A sequence of approximated solutions to the S-K model for spin glasses. J. Phys. A : Math. Gen. 13 115-121. |

[17] | Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin-glass. Phys. Rev. Lett. 35 1792-1796. |

[18] | Sudakov, V. N. and Cirel’son, B. S. (1974). Extremal properties of half-spaces for spherically invariant measures. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. ( LOMI ) 41 14-24, 165. Problems in the theory of probability distributions, II. · Zbl 0395.28007 |

[19] | Talagrand, M. (1987). Regularity of Gaussian processes. Acta Math. 159 99-149. · Zbl 0712.60044 |

[20] | Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221-263. · Zbl 1137.82010 |

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