Gaussian Processes and Local Times of Symmetric Lévy Processes

Marcus, Michael B.; Rosen, Jay

doi:10.1007/978-1-4612-0197-7_4

Cited by 12 publications

(9 citation statements)

References 6 publications

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“…The distribution of the local times for a Borel right process can be fully characterized by certain associated Gaussian processes; results of this flavor go by the name of Isomorphism Theorems. Several versions have been developed by Ray [44] and Knight [33], Dynkin [18,17], Marcus and Rosen [40,41], Eisenbaum [19] and Eisenbaum, Kaspi, Marcus, Rosen and Shi [20]. In what follows, we present the second Ray-Knight theorem in the special case of a continuous-time random walk.…”

Section: Outlinementioning

confidence: 99%

Cover times, blanket times, and majorizing measures

2012

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We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n)2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

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Section: Outlinementioning

confidence: 99%

Cover times, blanket times, and majorizing measures

2012

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“…The distribution of the local times for a Borel right process can be fully characterized by a certain associated Gaussian processes; results of this flavor go by the name of Dynkin isomorphism theory. Several versions have been developed by Ray [40] and Knight [30], Dynkin [21,22], Marcus and Rosen [35,36], Eisenbaum [23] and Eisenbaum et al [24]. In what follows, we present the second Ray-Knight theorem in the special case of a continuous-time random walk.…”

Section: Preliminaries Electric Networkmentioning

confidence: 99%

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

Ding¹

2014

Ann. Probab.

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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.

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Section: Lemma 113 (Matthews Bound) For Everymentioning

confidence: 99%

Cover times, blanket times, and majorizing measures

Ding

Lee

Peres

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

115

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We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G.This allows us to resolve a number of open questions. We give a deterministic polynomialtime algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996), showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n)2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000).

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Gaussian Processes and Local Times of Symmetric Lévy Processes

Cited by 12 publications

References 6 publications

Cover times, blanket times, and majorizing measures

Cover times, blanket times, and majorizing measures

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

Cover times, blanket times, and majorizing measures

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