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Tree Polymers in the Infinite Volume Limit at Critical Strong Disorder

Part of:Special processesApplications to specific types of physical systemsStochastic processes

Published online by Cambridge University Press:  14 July 2016

Torrey Johnson and
Edward C. Waymire
Torrey Johnson*
Affiliation:
Oregon State University
Edward C. Waymire*
Affiliation:
Oregon State University
*
∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA.
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Abstract

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The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

Keywords

Multiplicative cascadeT-martingaletree polymerstrong disorder

MSC classification

Secondary:60K35: Interacting random processes; statistical mechanics type models; percolation theory60G42: Martingales with discrete parameter82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011, pp. 885 - 891
Copyright
Copyright © Applied Probability Trust 2011 

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