  We use a large single particle tracking data set to analyze the short time
and small spatial scale motion of quantum dots labeling proteins in cell
membranes. Our analysis focuses on the jumps which are the changes in the
position of the quantum dots between frames in a movie of their motion.
Previously we have shown that the directions of the jumps are uniformly
distributed and the jump lengths can be characterized by a double power law
distribution.
  Here we show that the jumps over a small number of time steps can be
described by scalings of a {\em single} double power law distribution. This
provides additional strong evidence that the double power law provides an
accurate description of the fine scale motion. This more extensive analysis
provides strong evidence that the double power law is a novel stable
distribution for the motion. This analysis provides strong evidence that an
earlier result that the motion can be modeled as diffusion in a space of
fractional dimension roughly 3/2 is correct. The form of the power law
distribution quantifies the excess of short jumps in the data and provides an
accurate characterization of the fine scale diffusion and, in fact, this
distribution gives an accurate description of the jump lengths up to a few
hundred nanometers. Our results complement of the usual mean squared
displacement analysis used to study diffusion at larger scales where the
proteins are more likely to strongly interact with larger membrane structures.
