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Refined Questions

  • Wolfgang König
Chapter
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Part of the Pathways in Mathematics book series (PATHMATH)

Abstract

Let us survey a number of questions around the PAM that go beyond the basic questions that we have treated so far. In Sect. 7.1 we show what refined techniques can say about deeper analysis of the moment asymptotics of the total mass. Correlated potentials are considered in Sect. 7.2. In Sect. 7.3, we multiply the potential with a small t-dependent prefactor and examine how the concentrated behaviour is turned into some homogenised one. In Sects. 7.4 and 7.5 we discuss connections between the research on the PAM and on the upper tails of the random walk in random scenery and of general self-attractive functionals of the local times, respectively. In Sect. 7.6 we make a few remarks on general self-attractive path measures, in Sect. 7.7 we show how to interpolate between the moment asymptotics and the almost-sure asymptotics of the total mass, in Sect. 7.8 we report on research on the PAM with other random paths than the simple random walk. Results for the PAM in random environment (i.e., when the simple random walk is replaced by some random walk in random environment) are described in Sect. 7.9. In Sect. 7.10 we briefly characterize the relationship of the PAM with another model of high interest, the directed polymers in random environment, and in Sect. 7.11 we discuss some recent research on branching random walks in random environment that was inspired by the research on the PAM.

Keywords

Random Walk Random Environment Random Potential Simple Random Walk Critical Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Wolfgang König
    • 1
    • 2
  1. 1.für Angewandte Analysis und StochastikWeierstraß-InstitutBerlinGermany
  2. 2.Institute for MathematicsTU BerlinBerlinGermany

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