Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage.

Automated summary

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements previous studies restricted to dimension three and higher, adding to the research on random walk range and the volume of the Wiener sausage.