  We provide Monte Carlo estimates of the scaling of the length $L_{n}$ 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, $L_{n} \sim n^{\theta}$ with
the leading scaling exponent $0.60 \lesssim \theta \lesssim 0.69$ for the
heavy-tailed distributions of step lengths examined, with values increasing as
the distribution becomes more heavy-tailed, and $\theta \simeq 0.57$ for
distributions of finite variance, irrespective of the particular distribution.
The results are consistent with existing rigorous bounds for $\theta$, although
in a somewhat surprising manner. For random walks with step lengths of finite
variance, we conjecture that the correct asymptotic behavior of $L_{n}$ is
given by $\sqrt{n}\ln n$, and also propose the form of the subleading
asymptotics. The distribution of $L_{n}$ was found to follow a simple scaling
form with scaling functions that vary with $\theta$. 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.
