Voids in a $\Lambda$CDM Universe
  We study the formation and evolution of voids in the dark matter distribution using various simulations of the popular $\Lambda$ Cold Dark Matter cosmogony. We identify voids by requiring them to be regions of space with a mean overdensity of -0.8 or less. Each of the simulations contains thousands of voids. The distribution of void sizes in the different simulations shows good agreement. Voids very clearly correspond to minima in the smoothed initial density field. We find a universal void mass profile of the form $\rho(<r)/\rho(r_{\rm eff}) \propto \exp[(r/r_{\rm eff})^\alpha]$ where $r_{\rm eff}$ is the effective radius of a void and $\alpha\sim 2$. The mass function of haloes in voids is steeper than that of haloes that populate denser regions. The abundances of void haloes seem to evolve somewhat more strongly between redshifts $\sim 1$ and 0 than the global abundances of haloes. (abridged) 
Joerg M. Colberg, Ravi K. Sheth, Antonaldo Diaferio, Liang Gao, Naoki   Yoshida