  In glass forming liquids close to the glass transition point, even a very
slight increase in the macroscopic density results in a dramatic slowing down
of the macroscopic relaxation. Concomitantly, the local density itself
fluctuates in space. Therefore, one can imagine that even very small local
density variations control the local glassy nature. Based on this perspective,
a model for describing growing length scale accompanying the vitrification is
introduced, in which we assume that in a subsystem whose density is above a
certain threshold value, $\rho_{\rm c}$, owing to steric constraints, particle
rearrangements are highly suppressed for a sufficiently long time period
($\sim$ structural relaxation time). We regard such a subsystem as a glassy
cluster. Then, based on the statistics of the subsystem-density, we predict
that with compression (increasing average density $\rho$) at a fixed
temperature $T$ in supercooled states, the characteristic length of the
clusters, $\xi$, diverges as $\xi\sim(\rho_{\rm c}-\rho)^{-2/d}$, where $d$ is
the spatial dimensionality. This $\xi$ measures the average persistence length
of the steric constraints in blocking the rearrangement motions and is
determined by the subsystem density. Additionally, with decreasing $T$ at a
fixed $\rho$, the length scale diverges in the same manner as $\xi\sim(T-T_{\rm
c})^{-2/d}$, for which $\rho$ is identical to $\rho_{\rm c}$ at $T=T_{\rm c}$.
The exponent describing the diverging length scale is the same as the one
predicted by some theoretical models and indeed has been observed in some
simulations and experiments. However, the basic mechanism for this divergence
is different; that is, we do not invoke thermodynamic anomalies associated with
the thermodynamic phase transition as the origin of the growing length scale.
We further present arguements for the cooperative properties based on the
clusters.
