Inference of a random potential from random walk realizations: Formalism and application to the one-dimensional Sinai model with a drift

We consider the Sinai model, in which a random walker moves in a random quenched potential V, and ask the following questions: 1. how can the quenched potential V be inferred from the observations of one or more realizations of the random motion? 2. how many observations (walks) are required to make a reliable inference, that is, to be able to distinguish between two similar but distinct potentials, V₁ and V₂? We show how question 1 can be easily solved within the Bayesian framework. In addition, we show that the answer to question 2 is, in general, intimately connected to the calculation of the survival probability of a fictitious walker in a potential W defined from V₁ and V₂, with partial absorption at sites where V ₁ and V₂ do not coincide. For the one-dimensional Sinai model, this survival probability can be analytically calculated, in excellent agreement with numerical simulations.