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In the present work, we investigate the case of Directed Polymer in a Random Environment (DPRE), when the increments of the random walk are heavy-tailed with tail-exponent equal to zero ($\mathbf{P}[|X_1|\geq n]$ decays slower than any power of $n$).
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Wei, R.: On the long-range directed polymer model. J. Stat. Phys. 165(2), 320–350 (2016)
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K. Alexander, N. Zygouras. Equality of critical points for polymer depinning transitions with loop exponent one
2010
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H. Lacoin New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2
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S. Purkayastha Simple proofs of two results on convolutions of unimodal distributions
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J. Sohier Hierarchical pinning model: low disorder relevance in the b=s case
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E. Bates, Localization of directed polymers with general reference walk
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