Large deviations of the range of the planar random walk on the scale of the mean

Abstract

We show an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane with finite sixth moment. This result complements the study of Van den Berg, Bolthausen and Den Hollander, where the continuum case of the Wiener Sausage is studied, and in Phetpradap, in which one is restricted to dimension three and higher.