The Annals of Applied Probability

Distribution of levels in high-dimensional random landscapes

Zakhar Kabluchko

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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.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 1 (2012), 337-362.

Dates
First available in Project Euclid: 7 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1328623702

Digital Object Identifier
doi:10.1214/11-AAP772

Mathematical Reviews number (MathSciNet)
MR2932549

Zentralblatt MATH identifier
1246.60036

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Central limit theorem empirical process disordered systems long-range dependence Hermite polynomials reduction principle

Citation

Kabluchko, Zakhar. Distribution of levels in high-dimensional random landscapes. Ann. Appl. Probab. 22 (2012), no. 1, 337--362. doi:10.1214/11-AAP772. https://projecteuclid.org/euclid.aoap/1328623702


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