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Title: Smooth SLE
Author: Harper, J. M.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2007
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The search for the scaling limits of two-dimensional critical systems has been a very active area of research in Statistical Physics. Recently, a new technique for the analysis of these scaling limits was developed after the introduction of a process known as the Stochastic Loewner Evolution (SLE) by Oded Schramm. This process is a stochastic version of the Loewner chain from classical complex analysis, in which the driving function of the chain is taken to be a Brownian motion. The behaviour of SLE has been widely studied and has enabled the rigorous identification of the scaling limits of several discrete critical systems including critical site percolation on the triangular lattice. Many other systems are also conjectured to converge to some SLE-related process. The study of the behaviour of SLE has also lead to the exact calculation of many quantities relating to the asymptotic behaviour of these discrete models. In this study, we review the theory of classical Loewner chains before introducing a process that can be viewed as a smooth analogue of a Loewner chain. We study the properties of this process and show that it possesses a greater degree of regularity than a classical Loewner chain; in particular, we show that the planar domains associated with the smooth chain have smooth boundaries. We then study these smooth chains in the stochastic setting of Schramm, calling the resulting process a smooth SLE. One main difference between SLE and smooth SLE is that SLE possesses a certain scale-invariance, whereas smooth SLE processes do not. This scale-invariance is often crucial to the proofs of the behaviour of SLE and therefore any proof of corresponding behaviour for the smooth case must use other methods. The extra regularity of the smooth chain implies that some of the behavioural exhibited by an SLE process is not seen for a smooth chain. However, the main behavioural phase change, seen for SLE when the variance parameter of the driving Brownian motion is changed, is also seen in the smooth case. Finally, we examine issues relating to the approximation of smooth Loewner chains and also the convergence of these smooth chains to their classical counterparts.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available