  We investigate the statistics of three kinds of records associated with
planar random walks, namely diagonal, simultaneous and radial records. The mean
numbers of these records grow as universal power laws of time, with respective
exponents 1/4, 1/3 and 1/2. The study of diagonal and simultaneous records
relies on the underlying renewal structure of the successive hitting times and
locations of translated copies of a fixed target. In this sense, this work
represents a two-dimensional extension of the analysis made by Feller of ladder
points, i.e., records for one-dimensional random walks. This approach yields a
variety of analytical asymptotic results, including the full statistics of the
numbers of diagonal and simultaneous records, the joint law of the epoch and
location of the current diagonal record and the angular distribution of the
current simultaneous record. The sequence of radial records cannot be
constructed in terms of a renewal process. In spite of this, their mean number
is shown to grow with a super-universal square-root law for isotropic random
walks in any spatial dimension. Their full distribution is also obtained.
Higher-dimensional diagonal and simultaneous records are also briefly
discussed.
