Precise Asymptotics for a Random Walker's Maximum

Abstract

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/σ=\sqrt{2n/π}+ γ+O(n^{-1/2}), where σ^2 is the variance of the step lengths. While the leading \sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant γ. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed γ=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for γvalid for arbitrary symmetric distributions. We also demonstrate how γappears in the thermodynamic limit as the leading behavior of the difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to Lévy walks.

new references added, typos corrected, published version

References in corpus (4)

• Exact Maximal Height Distribution of Fluctuating Interfaces
• Maximal Height Scaling of Kinetically Growing Surfaces
• On the area under a continuous time Brownian motion till its first-passage time
• Statistics of extremal intensities for Gaussian interfaces

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