Abstract
In this chapter we start to analyse the behavior of the polymer in its localized regime. We will make precise the image of the corridors where the polymer wants to be. We will see that localization happens at all temperature in dimension d = 1 and 2. We illustrate the phenomenon by simulation experiments. Finally, a precise picture will be achieved when the environment has heavy tails. However, we leave some matter on the localized phase for the forthcoming Sects. 7.4 and 9.7
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Notes
- 1.
This will ensure uniqueness of maximizers.
- 2.
The reader should recall at this point that we are considering a model where the temperature is not kept constant, but rescaled with n.
- 3.
U n 1 > U n 2 > … are the values of ω in the quadrant \(\{(t,x) \in (n\mathcal{D})\bigcap (\mathbb{N} \times \mathbb{Z}),t \leftrightarrow x\}\) in decreasing order, and the \(Z_{n}^{i} \in \vert \![1,n]\!\vert \times \mathbb{Z}\) are the corresponding locations in the quadrant.
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Comets, F. (2017). The Localized Phase. In: Directed Polymers in Random Environments. Lecture Notes in Mathematics(), vol 2175. Springer, Cham. https://doi.org/10.1007/978-3-319-50487-2_6
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