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• W3100144060 endingPage "P06013" @default.
• W3100144060 startingPage "P06013" @default.
• W3100144060 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]/sigma=sqrt{2n/pi}+ gamma +O(n^{-1/2}), where sigma^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 gamma. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed gamma=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for gamma valid for arbitrary symmetric distributions. We also demonstrate how gamma appears 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 L'evy walks." @default.
• W3100144060 created "2020-11-23" @default.
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• W3100144060 date "2005-06-20" @default.
• W3100144060 modified "2023-10-17" @default.
• W3100144060 title "Precise asymptotics for a random walker’s maximum" @default.
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• W3100144060 doi "https://doi.org/10.1088/1742-5468/2005/06/p06013" @default.
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