The random average process and random walk in a space-time random environment in one dimension. (English) Zbl 1129.60097

This paper studies the so-called Random Average Process (RAP) using techniques that involve limiting behaviors of the quenched-mean process for a random walk in a space-time random environment. RAP processes design a class of (non-diffusive) 1d-asymmetric interacting systems whose random evolution produces fluctuations of order \(n^{1/4}\). It roughly speaking describes an interface between two phases on the plane. More formally, a real-valued function called, the height function, evolves on the integers by jumping to random convex combinations of its neighbors.
On the contrary to previous works on the subject, the results stated for the RAP in this paper concern the general hydrodynamic limit framework, modulo a few assumptions on the initial increments of the random height function. Its analysis uses a dual description in terms of backward random walks in a space-time environment, that fluctuates also at the scale of \(n^{1/4}\). At this scale, the limits of the fluctuations are described by a family of Gaussian processes and in the case of known product-form invariant distributions, by a two-parameter process whose time marginals consist of fractional Brownian motions with Hurst parameter \(1/4\). Tightness at the level of processes is left to the reader, except in the central case of the forward quenched means of random walks, for which the authors achieve it by computing a bound on the 6th moment of the process environment.
Related results for other sub-diffusive processes, e.g., the Hammersley process, are also reviewed in this paper together with discussions around some universality questions.


60K37 Processes in random environments
82C22 Interacting particle systems in time-dependent statistical mechanics
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