We investigate the statistics of the gap $G_n$ between the two rightmost positions of a Markovian one-dimensional random walker (RW) after $n$ time steps and of the duration $L_n$ which separates the occurrence of these two extremal positions. The distribution of the jumps ${\eta }_i$’s of the RW, $f(\eta )$, is symmetric and its Fourier transform has the small $k$ behavior $1-\widehat{f}(k)\sim |k{|}^{\mu }$, with $0<\mu \le 2$. For $\mu =2$, the RW converges, for large $n$, to Brownian motion, while for $0<\mu <2$ it corresponds to a Lévy flight of index $\mu$. We compute the joint probability density function (PDF) $P_n(g,l)$ of $G_n$ and $L_n$ and show that, when $n\to \infty$, it approaches a limiting PDF $p(g,l)$. The corresponding marginal PDFs of the gap, $p_{\mathrm{gap}}(g)$, and of $L_n$, $p_{\mathrm{time}}(l)$, are found to behave like $p_{\mathrm{gap}}(g)\sim g^{-1-\mu }$ for $g\gg 1$ and $0<\mu <2$, and $p_{\mathrm{time}}(l)\sim l^{-\gamma (\mu )}$ for $l\gg 1$ with $\gamma (1<\mu \le 2)=1+1/\mu$ and $\gamma (0<\mu <1)=2$. For $l$, $g\gg 1$ with fixed $l{g}^{-\mu }$, $p(g,l)$ takes the scaling form $p(g,l)\sim g^{-1-2\mu }{\tilde{p}}_{\mu }(l{g}^{-\mu })$, where ${\tilde{p}}_{\mu }(y)$ is a ($\mu$-dependent) scaling function. We also present numerical simulations which verify our analytic results.

• Received 21 March 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.070601