On the Gap and Time Interval between the First Two Maxima of Long Random Walks

52 pages, 8 figures. Published version (typos corrected).International audienceIn the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker, starting from $x_0 = 0$, after $n$ time steps (taking the $x$-axis vertical). The jumps $\eta_i = x_i - x_{i-1}$ are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), $f(\eta)$, the Fourier transform of which has the small $k$ behavior $1 - \hat f(k) \propto |k|^\mu$, with $0 < \mu \leq 2$. For $\mu=2$, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For $0<\mu