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The localization transition for the directed polymer in a random environment is smooth
Hubert Lacoin1 orcid
1IMPA Estrada Dona Castorina 110, Rio de Janeiro RJ-22460-320, Brasil
Publications Mathématiques de l'IHÉS, Online first, pp. 1-27
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When $d \ge 3$, the directed polymer in a random environment on $\mathbb{Z}^{d}$ displays a phase transition from a diffusive phase, known as weak disorder, to a localized phase, referred to as strong disorder. This transition is encoded by the behavior of the free energy of the model, defined by

\[ \mathfrak{f}(\beta ) := \lim _{N \rightarrow \infty } (1/n)\log W^{\beta }_n, \]

where $W^{\beta }_n$ is the normalized partition function for the directed polymer of length $n$. More precisely, weak disorder corresponds to $\mathfrak{f}(\beta )=0$ and strong disorder to $\mathfrak{f}(\beta )<0$. Monotonicity and continuity of $\mathfrak{f}$ imply that there exists $\beta _c \in [0, \infty ]$ such that weak disorder is equivalent to $\beta \in [0,\beta _c]$. Furthermore, $\beta _c>0$ if and only if $d \ge 3$. We prove that this transition is infinitely smooth in the sense that $\mathfrak{f}$ grows slower than any power function at the vicinity of $\beta _c$, that is

\[ \lim _{\beta \downarrow \beta _c}\frac{\log \vert {\mathfrak{f}(\beta )}\vert }{\log (\beta -\beta _c)}= \infty . \]

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Accepted:
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DOI: 10.5802/pmihes.28
Classification: 60K35, 60K37, 82B26, 82B27, 82B44
Keywords: directed polymers, phase transition, disordered system, essential singularity

Hubert Lacoin  1

1 IMPA Estrada Dona Castorina 110, Rio de Janeiro RJ-22460-320, Brasil 
Hubert Lacoin. The localization transition for the directed polymer in a random environment is smooth. Publications Mathématiques de l'IHÉS, Online first, pp. 1-27
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