Self-repelling walk on the Sierpiński gasket
The authors study a family, parametrized by \(u \in [0,1]\), of self-repelling processes constructed by taking continuum limits of the sequences of random walks on the graph approximations to the (finite) Sierpiński gasket. The self-similar processes corresponding to \(u \in (0,1)\) interpolate between the self-avoiding process (case \(u=0\)) and the Brownian motion (\(u=1\)), so that the path measure \(P^u\) is weakly continuous in \(u\) and the order of the Hölder continuity of the sample path is also continuous in the parameter. In contrast to some previous models, that originated in the studies of polymers, the parameter does not directly count the number of returns of the walks to bonds or sites. This new construction can produce different interpolating families of self-similar processes on \(\mathbb{R}\) when applied to random walks on \(\mathbb{Z}\). Moreover, a law of the iterated logarithm is established for thus arising processes via the large deviations estimates for the supercritical branching process describing the path.
scientific article; zbMATH DE number 1837437