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Random Loewner Chains in Riemann Surfaces


Zhan, Dapeng (2004) Random Loewner Chains in Riemann Surfaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/TK0N-3F57.


The thesis describes an extension of O. Schramm's SLE processes to complicated plane domains and Riemann surfaces. First, three kinds of new SLEs are defined for simple conformal types. They have properties similar to traditional SLEs. Then harmonic random Loewner chains (HRLC) are defined in finite Riemann surfaces. They are measures on the space of Loewner chains, which are increasing families of closed subsets satisfying certain properties. An HRLC is first defined on local charts using Loewner's equation. Since the definitions in different charts agree with each other, these local HRLCs can be put together to construct a global HRLC. An HRLC in a plane domain can be described by differential equations involving canonical plane domains. Those old and new SLEs are special cases of HRLCs. An HRLC is determined by a parameter K ≥ 0, a starting point and a target set. When K = 6, the HRLC satisfies the locality property. When K = 2, the HRLC preserves some observable that resembles the observable for the corresponding loop-erased random walk (LERW). So HRLC₂ should be the scaling limit of LERW. With reasonable assumptions, HRLC8/3 differs from a restriction measure by a conformally invariant density; for K ∈ (0,8/3), HRLCK differs from a pre-restriction measure by a conformally invariant density. A restriction measure could be constructed from a pre-restriction measure by adding Brownian bubbles.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Loewner chain; Riemann surfaces; SLE
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2004.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Makarov, Nikolai G.
Thesis Committee:
  • Makarov, Nikolai G. (chair)
  • Borodin, Alexei
  • Simon, Barry M.
  • Berger, Noam
Defense Date:24 May 2004
Non-Caltech Author Email:dapeng_zhan (AT)
Record Number:CaltechETD:etd-08102004-142550
Persistent URL:
Zhan, Dapeng0000-0001-5528-8142
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3079
Deposited By: Imported from ETD-db
Deposited On:11 Aug 2004
Last Modified:20 Jan 2021 23:51

Thesis Files

PDF (D_Zhan_thesis.pdf) - Final Version
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