  We consider a one-dimensional random walk (RW) with a continuous and
symmetric jump distribution, $f(\eta)$, characterized by a L\'evy index $\mu
\in (0,2]$, which includes standard random walks ($\mu=2$) and L\'evy flights
($0<\mu<2$). We study the survival probability, $q(x_0,n)$, representing the
probability that the RW stays non-negative up to step $n$, starting initially
at $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for
large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$
varies: (i) for $x_0= O(1)$ ("quantum regime"), the discreteness of the jump
process significantly alters the standard scaling behavior of $q(x_0,n)$ and
(ii) for $x_0 = O(n^{1/\mu})$ ("classical regime") the discrete-time nature of
the process is irrelevant and one recovers the standard scaling behavior (for
$\mu =2$ this corresponds to the standard Brownian scaling limit). The purpose
of this paper is to study how precisely the crossover in $q(x_0,n)$ occurs
between the quantum and the classical regime as one increases $x_0$.
