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A L\'evy random medium, in a given space, is a random point process where thedistances between points, a.k.a. targets, are long-tailed. Random walksvisiting the targets of a L\'evy random medium have been used to model many(physical, ecological, social) phenomena that exhibit superdiffusion as theresult of interactions between an agent and a sparse, complex environment. Inthis note we consider the simplest non-trivial L\'evy random medium, a sequenceof points in the real line with i.i.d. long-tailed distances betweenconsecutive targets. A popular example of a continuous-time random walk in thismedium is the so-called L\'evy-Lorentz gas. We give an account of a number ofrecent theorems on generalizations and variations of such model, in discreteand continuous time.

Discrete- and continuous-time random walks in 1D L\'evy random medium