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February 2012 Distribution of levels in high-dimensional random landscapes
Zakhar Kabluchko
Ann. Appl. Probab. 22(1): 337-362 (February 2012). DOI: 10.1214/11-AAP772

Abstract

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to ∞. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington–Kirkpatrick and Edwards–Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.

Citation

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Zakhar Kabluchko. "Distribution of levels in high-dimensional random landscapes." Ann. Appl. Probab. 22 (1) 337 - 362, February 2012. https://doi.org/10.1214/11-AAP772

Information

Published: February 2012
First available in Project Euclid: 7 February 2012

zbMATH: 1246.60036
MathSciNet: MR2932549
Digital Object Identifier: 10.1214/11-AAP772

Subjects:
Primary: 60F05
Secondary: 60K35

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.22 • No. 1 • February 2012
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