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On the range of the transient frog model on ℤ

Part of: Special processes Markov processes Basic hypergeometric functions Distribution theory - Probability

Published online by Cambridge University Press:  26 June 2017

Arka Ghosh ,
Steven Noren  and
Alexander Roitershtein
Arka Ghosh*
Affiliation:
Iowa State University
Steven Noren*
Affiliation:
Iowa State University
Alexander Roitershtein*
Affiliation:
Iowa State University
*
* Postal address: Department of Statistics, Iowa State University, Ames, IA 50011, USA.
** Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
** Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
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Abstract

We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

Keywords

Frog modelinteracting random walkrandom walks rangeq-series

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60E05: Distributions 33D05: $q$-gamma functions, $q$-beta functions and integrals
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 327 - 343
Copyright
Copyright © Applied Probability Trust 2017 

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