M.Sc Student | Louidor Oren |
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Subject | Pinning Phenomena and Models of Directed Polymers |

Department | Department of Industrial Engineering and Management |

Supervisor | Professor Dmitry Ioffe |

Full Thesis text |

We consider three independent problems involving Directed Random Walks or Polymers in Z^{2}, emerging from the study of
several models of Disordered Media, including the
Directed Polymers In Random Environment (DPRE) Model, the Polymer
Near An Attractive Wall (PNAW) Model and the Percolation Model. In one problem, we consider a simplified DPRE model, where the
environment has fixed valued sites along the path of an
additionally drawn directed RW, outside of which the
sites are zero. In this model we prove an almost surely quenched weak
localization of the polymer to the environment,
for all positive interaction strengths. In the second problem,
we consider a model similar to PNAW, only that the polymer’s RW is nonhomogenous, with steps having distributions chosen a
priori (deterministically) and satisfying some
weak restrictions. We prove here weak localization for all positive
interaction strengths, as well. The last problem involves estimating the probability that two RWs in Z^{2}, directed
along the first axis, and symmetric in the other, satisfying some
weak restrictions, starting from the origin, reach point (N, 0) without
their interpolated paths intersecting before. We show an asymptotical behavior of order N^{-2}. This result is useful
in deriving a precise formula for the Truncated Two- Point
Connectivity Function in the Z2 - Bond Percolation Model.