Current fluctuations are a topic of much interest in the area of interacting particle systems. In this thesis we study current fluctuations in a system of independent and identically distributed random walks. In this particle system, the hydrodynamic limit of particle density satisfies a linear first-order hyperbolic partial differential equation (i.e. the transport equation). The characteristics of this partial differential equation are straight lines with slope given by the average velocity of the random walks. In 2003, Seppalainen first studied current fluctuations across characteristics for a system of asymmetric independent random walks in one dimension, for a large class of initial configurations. In this thesis we extend his results and find the scaling limit of particle currents off characteristics. We construct a two parameter current process, the parameters being time and spatial distance from the characteristics, and show that it converges to a two parameter Gaussian process with given covariance. We also extend the idea of currents across characteristics to higher dimensions and construct a distribution valued current process to capture fluctuations in a system of independent random walks in multiple dimensions. The scaling limit of the current process in multiple dimensions is found to be a distribution valued Gaussian process with given covariance. Large deviation results about the current process in the one dimensional model are also presented.
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