Abstract
Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an $\alpha$-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.
Highlights
The probabilistic model of directed polymers in random environment was introduced by Imbrie and Spencer [34] as a reformulation of Huse and Henley’s approach [33] to studying the phase boundary of the Ising model in the presence of random impurities
In its classical form, the model considers a simple random walk (SRW) on the integer lattice Zd, whose paths—considered the “polymer”—are reweighted according to a random environment that refreshes at each time step
Lévy flights in random potentials have been used to study chemical reactions [19] and particle dispersions [57, 14]. Their continuous-time analogs, Lévy processes, appear in a variety of disciplines including fluid mechanics, solid state physics, polymer chemistry, and mathematical finance [8]. This is relevant because α-stable polymers are known to obey a scaling CLT at sufficiently high temperatures ([22, Theorem 4.2] and [66, Theorem 1.9]), which generalizes the Brownian CLT proved in [29, Theorem 1.2] when the reference walk is SRW
Summary
The probabilistic model of directed polymers in random environment was introduced by Imbrie and Spencer [34] as a reformulation of Huse and Henley’s approach [33] to studying the phase boundary of the Ising model in the presence of random impurities. Lévy flights in random potentials have been used to study chemical reactions [19] and particle dispersions [57, 14] Their continuous-time analogs, Lévy processes, appear in a variety of disciplines including fluid mechanics, solid state physics, polymer chemistry, and mathematical finance [8]. This is relevant because α-stable polymers are known to obey a scaling CLT at sufficiently high temperatures ([22, Theorem 4.2] and [66, Theorem 1.9]), which generalizes the Brownian CLT proved in [29, Theorem 1.2] when the reference walk is SRW.
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