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We consider the Nonlinear-Cost Random Walk model in discrete time introducedin [Phys. Rev. Lett. 130, 237102 (2023)], where a fee is charged for each jumpof the walker. The nonlinear cost function is such that slow/short jumps incura flat fee, while for fast/long jumps the cost is proportional to the distancecovered. In this paper we compute analytically the average and variance of thedistance covered in $n$ steps when the total budget $C$ is fixed, as well asthe statistics of the number of long/short jumps in a trajectory of length $n$,for the exponential jump distribution. These observables exhibit a very richand non-monotonic scaling behavior as a function of the variable $C/n$, whichis traced back to the makeup of a typical trajectory in terms of long/shortjumps, and the resulting "entropy" thereof. As a byproduct, we compute theasymptotic behavior of ratios of Kummer hypergeometric functions when both thefirst and last arguments are large. All our analytical results are corroboratedby numerical simulations.

Nonlinear-Cost Random Walk: exact statistics of the distance covered for fixed budget