Browsing by Author "Choi, Changsun"

This thesis presents a work on the hierarchical models related to Mathematical Physics. It first derives the space-time covariance of the solution of a linear parabolic stochastic partial differential equation (SPDE) with a space-time white noise, where the solution tends in distribution to the Gaussian Free Field (GFF) as the time goes to infinity. Next, it expresses the stationary measure for a non-linear SPDE of Ginzburg-Landau type as an integral of a derivative with respect to the GFF distribution. The rest of the thesis then focuses on searching for a possible Mathematical interpretation of the derivative, which in its turn involves an integral of the fourth power of a generalised random field. It is shown that the direct approach, a naive one using the averaging method or even the approach using Wick's power does not lead to a fully successful interpretation of the above integral, especially in the important case of dimension three. It therefore turns the attention to hierarchical models, first on the unit cube and later on the lattice points. The hierarchical model on the unit cube gives an exact relation between the Green function and the hierarchical free Gaussian random field as well as a satisfactory interpretation of our integral for dimension two, but not for three. On the other hand this model suggests that one could work with effective actions instead. The main part of the thesis lies on formulating a renormalisation group (RG) map for bounded functions and proving a convergence result for the iterations by computing cumulants. Namely, as the level, from which the iteration of RG starts and stops at a fixed level, tends to infinity, the iteration sequences converge. With the previous method in mind one translates the equations in the hierarchical model on the cube which correspond to the iteration of effective actions to those in the hierarchical model on the lattice points and obtains covariance estimates for the linearized part of the equation. These estimates can be used in establishing the convergence result for the effective actions.