The Annals of Probability

Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees

Jian Ding

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Abstract

In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.

Article information

Source
Ann. Probab., Volume 42, Number 2 (2014), 464-496.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251293

Digital Object Identifier
doi:10.1214/12-AOP822

Mathematical Reviews number (MathSciNet)
MR3178464

Zentralblatt MATH identifier
1316.60064

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G60: Random fields 60G15: Gaussian processes

Keywords
Cover times Gaussian free fields isomorphism theorem sprinkling method

Citation

Ding, Jian. Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees. Ann. Probab. 42 (2014), no. 2, 464--496. doi:10.1214/12-AOP822. https://projecteuclid.org/euclid.aop/1393251293


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