We study self avoiding random walks in an environment where sites are excluded randomly, in two and three dimensions. For a single polymer chain, we study the statistics of the time averaged monomer density and show that these are well described by multifractal statistics. This is true even far from the percolation transition of the disordered medium. We investigate solutions of chains in a disordered environment and show that the statistics cease to be multifractal beyond the screening length of the solution.
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