Self-repelling Brownian polymers at criticality

增长的聚合物链可以被建模为自避免的随机行走。我们研究了一类连续模型，Norris等人在1987年引入的自排斥布朗聚合物。在离散空间中也有类似的模型，即由Amit等人在1983年提出的近视自避免随机游走。在大于或等于3的维度中，这些模型很好地理解，并且在很大程度上可以用维纳过程来近似该过程。在两个维度中，临界维度，不能应用相同的方法。已知该过程在该临界尺寸中是超扩散的，但不知道精确的速率。我们的目的是调查这个模型的临界尺寸的属性，并试图使用最近的方法已被应用到不同的模型，以获得更精确的渐近速率的超扩散，即各向异性KPZ和扩散的高斯自由场的卷曲。作为一名校长，必须提出新的方法来分析临界维度中的现象，在临界维度中，经典方法将不起作用。在这个问题上的任何进展都将引起更广泛的社会的兴趣，因为它与研究临界状态下自相互作用随机现象的其他模型有关。

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.