  We extend the use of Configurational Information Measures (CIMSs) to
instantons and vacuum decay in arbitrary spatial dimensions. We find that both
the complexity and the information content in the shape of instanton solutions
have distinct regions of behavior in parameter space, discriminating between
qualitative thin and thick wall profiles. For Euclidean spaces of dimension
$D$, we show that for $D \gt 6$ instantons undergo a qualitative change of
behavior from their lower- dimensional counterparts, indicating that $D=6$ is a
critical dimension. We also find a scaling law relating the different CIMs to
the rate of vacuum decay, thus connecting the stability of the vacuum to the
informational complexity stored in the shape of the related critical bounce.
