We study space-time fluctuations around a characteristic line for aone-dimensional interacting system known as the random average process. Thestate of this system is a real-valued function on the integers. New values ofthe function are created by averaging previous values with random weights. Thefluctuations analyzed occur on the scale n^{1/4} where n is the ratio ofmacroscopic and microscopic scales in the system. The limits of thefluctuations are described by a family of Gaussian processes. In cases of knownproduct-form equilibria, this limit is a two-parameter process whose timemarginals are fractional Brownian motions with Hurst parameter 1/4. Along theway we study the limits of quenched mean processes for a random walk in aspace-time random environment. These limits also happen at scale n^{1/4} andare described by certain Gaussian processes that we identify. In particular,when we look at a backward quenched mean process, the limit process is thesolution of a stochastic heat equation.
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