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We consider a discrete-time random walk on a one-dimensional lattice withspace and time-dependent random jump probabilities, known as the Beta randomwalk. We are interested in the probability that, for a given realization of thejump probabilities (a sample), a walker starting at the origin at time $t=0$ isat position beyond $\xi \sqrt{T/2}$ at time $T$. This probability fluctuatesfrom sample to sample and we study the large-deviation rate function whichcharacterizes the tails of its distribution at large time $T \gg 1$. It isargued that, up to a simple rescaling, this rate function is identical to theone recently obtained exactly by two of the authors for the continuum versionof the model. That continuum model also appears in the macroscopic fluctuationtheory of a class of lattice gases, e.g. in the so-called KMP model of heattransfer. An extensive numerical simulation of the Beta random walk, based onan importance sampling algorithm, is found in good agreement with the detailedanalytical predictions. A first-order transition in the tilted measure,predicted to occur in the continuum model, is also observed in the numerics.

Probing the large deviations for the Beta random walk in random medium