  For a complete picture of galaxy cluster formation, it is important that we
start probing the early epoch of z~2-7 during which clusters and their galaxies
first began to form. Because the study of these so-called "proto-clusters" is
currently limited by small number statistics, widely varying selection
techniques and assumptions, we have performed a systematic study of cluster
formation utilizing cosmological simulations. We use the Millennium Simulations
to track the evolution of dark matter and galaxies in ~3,000 clusters from the
earliest times to z=0. We define an effective radius R_e for proto-clusters and
characterize their growth in size and mass. We show that the progenitor regions
of galaxy clusters (M>10^14 M_sun/h) can already be identified at least up to
z~5, provided that the galaxy overdensities, delta_gal, are measured on a
sufficiently large scale (R_e~5-10 cMpc). We present the overdensities in
matter, DM halos, and galaxies as functions of present-day cluster mass,
redshift, bias, and window size that can be used to interpret the structures
found in real surveys. We derive the probability that a structure having a
delta_gal, defined by a set of observational selection criteria, is indeed a
proto-cluster, and show how their z=0 masses can already be estimated long
before virialization. Galaxy overdensity profiles as a function of radius are
presented. We further show how their projected surface overdensities decrease
as the uncertainties in redshift measurements increase. We provide a table of
proto-cluster candidates selected from the literature, and discuss their
properties in the light of our simulations predictions. This work provides the
general framework that will allow us to extend the study of cluster formation
out to much higher redshifts using the large number of proto-clusters that are
expected to be discovered in, e.g., the upcoming HETDEX and HSC surveys.
