We provide Monte Carlo estimates of the scaling of the length Ln of the longest increasing subsequences of n-steps random walks for several different distributions of step lengths, short and heavy-tailed. Our simulations indicate that, barring possible logarithmic corrections, Ln∼nθ
Nearby in the stack
with the leading scaling exponent
0.60≲θ≲0.69
for the heavy-tailed distributions of step lengths examined, with values increasing as the distribution becomes more heavy-tailed, and
θ≃0.57
for distributions of finite variance, irrespective of the particular distribution. The results are consistent with existing rigorous bounds for
θ
, although in a somewhat surprising manner. For random walks with step lengths of finite variance, we conjecture that the correct asymptotic behavior of
Ln
is given by
nlnn
, and also propose the form of the subleading asymptotics. The distribution of
Ln
was found to follow a simple scaling form with scaling functions that vary with
θ
. Accordingly, when the step lengths are of finite variance they seem to be universal. The nature of this scaling remains unclear, since we lack a working model, microscopic or hydrodynamic, for the behavior of the length of the longest increasing subsequences of random walks.