  Consider "Frozen Random Walk" on $\mathbb{Z}$: $n$ particles start at the
origin. At any discrete time, the leftmost and rightmost
$\lfloor{\frac{n}{4}}\rfloor$ particles are "frozen" and do not move. The rest
of the particles in the "bulk" independently jump to the left and right
uniformly. The goal of this note is to understand the limit of this process
under scaling of mass and time. To this end we study the following
deterministic mass splitting process: start with mass $1$ at the origin. At
each step the extreme quarter mass on each side is "frozen". The remaining
"free" mass in the center evolves according to the discrete heat equation. We
establish diffusive behavior of this mass evolution and identify the scaling
limit under the assumption of its existence. It is natural to expect the limit
to be a truncated Gaussian. A naive guess for the truncation point might be the
$1/4$ quantile points on either side of the origin. We show that this is not
the case and it is in fact determined by the evolution of the second moment of
the mass distribution.
