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We consider a one-dimensional random walk (RW) with a continuous andsymmetric 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 theprobability that the RW stays non-negative up to step $n$, starting initiallyat $x_0 \geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ forlarge $n$. 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 jumpprocess 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 ofthe process is irrelevant and one recovers the standard scaling behavior (for$\mu =2$ this corresponds to the standard Brownian scaling limit). The purposeof this paper is to study how precisely the crossover in $q(x_0,n)$ occursbetween the quantum and the classical regime as one increases $x_0$.

Survival Probability of Random Walks and L\'evy Flights on a Semi-Infinite Line