Self-repelling Brownian polymers at criticality

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A growing polymer chain can be modelled as a self-avoiding random walk. We study a continuous model in this class, the self-repelling Brownian polymer introduced by Norris et al. in 1987. There is a similar model in discrete space, the myopic self-avoiding random walk introduced by Amit et al. in 1983. In dimensions greater than or equal to three, these models are well understood, and for large times the process can be approximated with a Wiener process. In two dimensions, the critical dimension, the same methods cannot be applied. It is known that the process is super-diffusive in this critical dimension, but the precise rate is not known. Our aim is to investigate the properties of this model in the critical dimension and attempt to use recent methods which have been applied to different models to obtain more precise asymptotic rates for the super-diffusivity, namely, anisotropic KPZ and a diffusion in the curl of the Gaussian free-field. As a principal, one must produce new methods to analyse phenomena in the critical dimension, where the classical methods will not work. Any progress on this problem will be of interest to the wider community, because of its relevance for studying other models for self-interacting random phenomena at criticality.