Precise asymptotics for a random walker’s maximum

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 , where σ 2 is the variance of the step lengths. While the leading behaviour 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 = −0.516 068... by exactly enumerating a lengthy double series. Here we present a closed exact formula for γ valid for arbitrary symmetric distributions. We also demonstrate how γ appears in the thermodynamic limit as the leading behaviour 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.