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2017 | OriginalPaper | Chapter

4. Lattice Versus Tree

Author : Francis Comets

Published in: Directed Polymers in Random Environments

Publisher: Springer International Publishing

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Abstract

In this chapter we deal with polymer models on different oriented graphs and compare them with the lattice case. As revealed by Derrida and Spohn in, many interesting questions can be answered on the regular tree. Later on, refined tree-like structures including Derrida’s m-tree has been introduced, yielding further comparisons. There, correlations are simpler compared to the lattice case, since the medium along two paths becomes independent as soon as they visit different sites. In the sense of simplifying the correlation structure, these models play the role of mean-field models.

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previous chapter The Martingale Approach and the L 2 Region
next chapter Semimartingale Approach and Localization Transition
Footnotes
1
Here, “large” means exponential in n, and “small” means subexponential—the bound on the entropy has been reached.
 
2
The proof below can be directly adapted to the case of a bounded, but otherwise general, environment η, by dividing the range of η(t, x) into three intervals, to take into account the variability of η. We will not further address this issue.
 
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Metadata
Title
Lattice Versus Tree
Author
Francis Comets
Copyright Year
2017
Book
Directed Polymers in Random Environments

Print ISBN: 978-3-319-50486-5

Electronic ISBN: 978-3-319-50487-2

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-50487-2_4