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ON THE DEPINNING TRANSITION OF THE DIRECTED POLYMER IN A
RANDOM ENVIRONMENT WITH A DEFECT LINE
by
G okhan Y ld r m
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
August 2013
Copyright 2013 G okhan Y ld r m
Object Description
| Title | On the depinning transition of the directed polymer in a random environment with a defect line |
| Author | Yıldırım, Gökhan; Yildirim, Gokhan |
| Author email | gkhnyildirim@gmail.com;gyildiri@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Mathematics |
| School | College of Letters, Arts And Sciences |
| Date defended/completed | 2013-06-05 |
| Date submitted | 2013-07-02 |
| Date approved | 2013-07-03 |
| Restricted until | 2013-07-03 |
| Date published | 2013-07-03 |
| Advisor (committee chair) | Alexander, Kenneth S. |
| Advisor (committee member) |
Baxendale, Peter H. Goldstein, Larry Haas, Stephan W. |
| Abstract | We study the depinning transition of the 1+1 dimensional directed polymer in a random environment with a defect line model. The monomer locations of the polymer is modeled by the space-time trajectory of a one dimensional simple symmetric random walk. Random environment is introduced by assigning each site of Z² an independent and identically distributed normal random variable that interacts with the polymer when it visits that site. The defect line is incorporated to the model by having an additional constant potential u at the origin which gives a reward or penalty to the polymer as it visits the origin. There is a critical value of u above which polymer is pinned, placing a positive fraction of its monomers at zero with high probability. ❧ Our first result is that the quenched free energy exists and the self-averaging holds, which implies that there is a nonrandom quenched critical point. To see the effect of disorder on the depinning transition, we compare the quenched free energy of the system as a function of u to the corresponding annealed system. Our main result is that the quenched and annealed free energies differ significantly only in a very small neighborhood of the critical point and we show that the size of this neighborhood scales as at most β as β → 0, where β is the inverse temperature. |
| Keyword | directed polymers; pinning models; disordered models; Lipschitz percolation |
| Language | English |
| Format (imt) | application/pdf |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Contributing entity | University of Southern California |
| Rights | Yıldırım, Gökhan; Yildirim, Gokhan |
| Physical access | The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-YldrmGghan-1735.pdf |
| Archival file | uscthesesreloadpub_Volume7/etd-YldrmGghan-1735.pdf |
Description
| Title | Page 1 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | ON THE DEPINNING TRANSITION OF THE DIRECTED POLYMER IN A RANDOM ENVIRONMENT WITH A DEFECT LINE by G okhan Y ld r m A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2013 Copyright 2013 G okhan Y ld r m |
