The Watershed Model and Schramm-loewner Evolution

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Daryaei, Ebrahim | 2013

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Type of Document: Ph.D. Dissertation
Language: Farsi
Document No: 44114 (04)
University: Sharif University of Technology
Department: Physics
Advisor(s): Rouhani, Shahin
Abstract:
Schramm Loewner evolution (SLE) is a one-parameter family of random simple curves in the complex plane introduced by Schramm in 1999 which is believed to describe the scaling limit of a variety of domain interfaces at criticality. This thesis is concerned with statistical properties of watersheds dividing drainage basins. The fractal dimension of this model is 1.22 which is consistent with the known fractal dimension for several important models such as Invasion percolation and minimum spanning trees (MST). We present numerical evidences that in the scaling limit this model are SLE curves with =1.73, being the only known physical example of an
SLE with <2. This lies outside the well-known duality conjecture. Finally we also numerically demonstrate all of these models are related to Loop erased random walk on critical percolation cluster which result in the same fractal dimension
Keywords:
Conformal Invariance ; Schramm-Loewner Evolution (SLE) ; Watershed Method ; Loop Erased Random Walk (LERW) ; Conformal Fields Theory