טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentLouidor Oren
SubjectPinning Phenomena and Models of Directed Polymers
DepartmentDepartment of Industrial Engineering and Management
Supervisor Professor Dmitry Ioffe
Full Thesis textFull thesis text - English Version


Abstract


We consider three independent problems involving Directed Random Walks or Polymers in Z2, 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 Z2, 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.