COMPLEXITY OF RANDOM SMOOTH FUNCTIONS ON THE HIGH-DIMENSIONAL SPHERE

被引:111
作者
Auffinger, Antonio [1 ]
Ben Arous, Gerard [2 ]
机构
[1] Univ Chicago, Chicago, IL 60637 USA
[2] NYU, Courant Inst Math Sci, New York, NY 10012 USA
基金
美国国家科学基金会;
关键词
Sample; spin glasses; critical points; random matrices; Parisi formula; RANDOM MATRICES;
D O I
10.1214/13-AOP862
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.
引用
收藏
页码:4214 / 4247
页数:34
相关论文
共 16 条
[1]  
Adler Robert J., 2007, Random Fields and Geometry, DOI DOI 10.1007/978-0-387-48116-6
[2]  
[Anonymous], 1998, LARGE DEVIATIONS TEC, DOI DOI 10.1007/978-1-4612-5320-4
[3]  
[Anonymous], 1942, Duke Math. J., DOI 10.1215/S0012-7094-42-00908-6
[4]   Random matrices and complexity of spin glasses [J].
Auffinger, Antonio ;
Ben Arous, Gerard ;
Cerny, Jiri .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2013, 66 (02) :165-201
[5]  
Azais J., 2009, Level Sets and Extrema of Random Processes and Fields, DOI [10.1002/9780470434642, DOI 10.1002/9780470434642]
[6]  
BenArous G, 1997, PROBAB THEORY REL, V108, P517
[7]  
Bleistein N., 2010, ASYMPTOTIC EXPANSION
[8]   Spherical 2+p spin-glass model:: An exactly solvable model for glass to spin-glass transition -: art. no. 217203 [J].
Crisanti, A ;
Leuzzi, L .
PHYSICAL REVIEW LETTERS, 2004, 93 (21)
[9]  
CRISANTI A, 1995, J PHYS I, V5, P805, DOI 10.1051/jp1:1995164
[10]   Replica symmetry breaking condition exposed by random matrix calculation of landscape complexity [J].
Fyodorov, Yan V. ;
Williams, Ian .
JOURNAL OF STATISTICAL PHYSICS, 2007, 129 (5-6) :1081-1116