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Decomposition of free fields and structural stability of dynamical systems for renormalization group analysis

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Title: Decomposition of free fields and structural stability of dynamical systems for renormalization group analysis
Author: Bauerschmidt, Roland
Degree Doctor of Philosophy - PhD
Program Mathematics
Copyright Date: 2013
Publicly Available in cIRcle 2013-08-15
Abstract: The main results of this thesis concern the spatial decomposition of Gaussian fields and the structural stability of a class of dynamical systems near a non-hyperbolic fixed point. These are two problems that arise in a renormalization group method for random fields and self-avoiding walks developed by Brydges and Slade. This renormalization group program is outlined in the introduction of this thesis with emphasis on the relevance of the problems studied subsequently. The first original result is a new and simple method to decompose the Green functions corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green function. This result gives rise to multiscale decompositions of the associated free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Such decompositions are the point of departure for renormalization group analysis. The novelty of the result is the use of the finite propagation speed of the wave equation and a related property of Chebyshev polynomials. The result improves several existing results and also gives simpler proofs. The second result concerns structural stability, with respect to contractive third-order perturbations, of a certain class of dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well- posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove the existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. This result can be applied to the renormalization group map of Brydges and Slade, and is an ingredient in the analysis of the long-distance behavior of four dimensional weakly self-avoiding walks using this approach.
URI: http://hdl.handle.net/2429/44817
Scholarly Level: Graduate

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