Overlaps and pathwise localization in the Anderson polymer model

Francis Comets and Michael Cranston

Stochastic Processes and their Applications, 2013, vol. 123, issue 6, 2446-2471

Abstract: We consider large time behaviour of typical paths under the Anderson polymer measure. If Pκx is the measure induced by rate κ, simple, symmetric random walk on Zd started at x, this measure is defined as dμκ,β,Tx(X)=Zκ,β,T(x)−1exp{β∫0TdWX(s)(s)}dPκx(X) where {Wx:x∈Zd} is a field of iid standard, one-dimensional Brownian motions, β>0,κ>0 and Zκ,β,T(x) the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighbourhood of a typical path as T→∞, for parameter values outside the perturbative regime of the random walk, giving a pathwise approach to polymer localization, in contrast with existing results. The localization becomes complete as β2κ→∞ in the sense that the mass grows to 1. The proof makes use of the overlap between two independent samples drawn under the Gibbs measure μκ,β,Tx, which can be estimated by the integration by parts formula for the Gaussian environment. Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling properties, thermodynamic limits, and decoupling of the parameters.

Keywords: Brownian polymer; Overlap; Malliavin calculus; Parabolic Anderson model (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414913000586
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:123:y:2013:i:6:p:2446-2471

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2013.02.010

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().