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Experimental Evidence of the Role of Compound Counting Processes in Random Walk Approaches to Fractional Dynamics

We present dielectric spectroscopy data obtained for gallium-doped Cd$_{0.99}$Mn$_{0.01}$Te:Ga mixed crystals which exhibit a very special case of the two-power-law relaxation pattern with the high-frequency power-law exponent equal to 1. We explain this behavior, which cannot be fitted by none of the well-known empirical relaxation functions, in a subordinated diffusive framework. We propose diffusion scenario based on a renormalized clustering of random number of spatio-temporal steps in the continuous time random walk. Such a construction substitutes the renewal counting process, used in the classical continuous time random walk methodology, by a compound counting one. As a result, we obtain a novel relaxation function governing the observed non-standard pattern, and we show the importance of the compound counting processes in studying fractional dynamics of complex systems.