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The asymptotic shape theorem for generalized first passage percolation
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2010 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, no 2, 632-660 p.Article in journal (Refereed) Published
Abstract [en]

We generalize the asymptotic shape theorem in first passage percolation on Z(d) to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for Z(d)-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon-Nikodym property.

Place, publisher, year, edition, pages
2010. Vol. 38, no 2, 632-660 p.
Keyword [en]
First passage percolation, cocycles, subadditive ergodic theory
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-13945DOI: 10.1214/09-AOP491ISI: 000275938300007Scopus ID: 2-s2.0-77953553228OAI: oai:DiVA.org:kth-13945DiVA: diva2:328522
Note
QC 20100705Available from: 2010-07-05 Created: 2010-07-05 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Limit Theorems for Ergodic Group Actions and Random Walks
Open this publication in new window or tab >>Limit Theorems for Ergodic Group Actions and Random Walks
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups.

In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd.

In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic.

Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vii, 18 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 09:06
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10451 (URN)
Public defence
2009-05-26, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100705Available from: 2009-05-18 Created: 2009-05-15 Last updated: 2012-03-27Bibliographically approved

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Citation style
  • apa
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