Statistics of Inflating Regions in Eternal Inflation
  We compute the distribution of sizes of inflating and non-inflating regions
in an eternally inflating Universe. As a first illustrative problem, we study a
simple scenario of an eternally inflating Universe in the presence of a
massless scalar field $\varphi$ whose field values lie within some finite
domain $\varphi\in(-\varphi_{cr},\varphi_{cr})$, with $\pm\varphi_{cr}$ marking
the onset of thermalization/crunching. We compute many important quantities,
including the fractal dimension, distribution of field values among inflating
regions, and the number of inflating and non-inflating Hubble regions. With the
aid of simulations in 1 spatial dimension, we show this eternally inflating
Universe reaches a steady state in which average sizes of inflating regions
grow only as a power law in the field's crunch value $\sim \varphi_{cr}^2$
(extension to higher dimensions is $\sim\varphi^{2/D}$), contrary to a naive
expectation of an exponential dependence. Furthermore, the distribution in
sizes exhibits an exponential fall off for large distances (with an initial
power law for inflating regions). We leave other interesting cases of more
realistic potentials and time varying Hubble parameter for future work, with a
possible application to the SM Higgs in the early Universe.
