  Random walk is a very important Markov process and has important applications
in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a
site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing
local time at time $n$ achieves the maximum among all sites. In this paper, we
study the cardinality of the favorite downcrossing site set, and will show that
with probability 1 there are only finitely many times at which there are at
least four favorite downcrossing sites and three favorite downcrossing sites
occurs infinitely often. Some related open questions will be introduced.
