Probabilistic analysis of directed polymers in a random environment: a review

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Introduction to KPZ (2011, background): “The fact that it is strict was proved in d = 1 by Comets et al [CSY04] (see also [Lac09]).”
Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension (2012, background): “On the other hand, the behavior of the polymer paths in strong disorder is different since there exist (random) points at which the path has a uniform (in n) positive probability (under dPˇ;Q) of ending (see [20, 21]).”
The Kardar-Parisi-Zhang equation and universality class (2011, background): “On the other hand, in strong disorder it is known (see [44]) that there exist (random) points at which the path π has a positive probability (under dP β Q) of ending.”
On the (strict) positivity of solutions of the stochastic heat equation (2012, background): “The SHE arises as the limit of the renormalized partition function of discrete directed polymers [1] and the CDP as the weak limit of the discrete directed polymer path measure (see [4] for a general review on directed polymers).”
Macdonald processes (2011, background): “On the other hand, in strong disorder it is known (see [28]) that there exist (random) points at which the path π has a positive probability (under dP β Q) of ending.”
Directed polymers in random environment are diffusive at weak disorder (2004, background): “Further motivations for the model can be found in the physics literature [14], [23], and a rigorous survey in [10].”