  For the initial fields of the density contrast and peculiar velocity, we
theoretically calculate the differential and integral length scales, i.e.,
statistical measures that respectively characterize the small- and large-scale
fluctuations of a random field. These length scales and the associated mass
scales explain the length and mass scales observed for (1) halos of young
galaxies at z > 5, (2) halos of galaxies at z = 0, and (3) the largest
structures in the galaxy distribution at z = 0. We thereby discuss that such
observed scales are fossil imprints of the characteristic scales of the initial
fields.
