Open Access
March 2010 The asymptotic shape theorem for generalized first passage percolation
Michael Björklund
Ann. Probab. 38(2): 632-660 (March 2010). DOI: 10.1214/09-AOP491


We generalize the asymptotic shape theorem in first passage percolation on ℤd 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 ℤd-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.


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Michael Björklund. "The asymptotic shape theorem for generalized first passage percolation." Ann. Probab. 38 (2) 632 - 660, March 2010.


Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1194.47013
MathSciNet: MR2642888
Digital Object Identifier: 10.1214/09-AOP491

Primary: 47A35 , 82B43

Keywords: cocycles , first passage percolation , subadditive ergodic theory

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 2 • March 2010
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