An application of extreme value statistics to the most massive galaxy
  clusters at low and high redshifts
  In this work we present an application of general extreme value statistics
(GEV) to very massive single clusters at high and low redshifts. After
introducing the formalism, we apply this statistics to four very massive high
redshift clusters. Those clusters comprise ACT-CL J0102-4915 with a mass of
M_200m=(2.16+/-0.32)x10^{15} M_sun at a redshift of z=0.87, SPT-CL J2106-5844
with a mass of M_200m=(1.27+/-0.21)x10^{15} M_sun at z=1.132 and two clusters
found by the XMM-Newton Distant Cluster Project survey: XMMU J2235.32557 with a
mass of M_200c= (7.3+/-1.3)x10^{14} M_sun located at a redshift of z=1.4 and
XMMU J0044.0-2033 having a mass in the range of M_200c= (3.5-5.0)x10^{14} M_sun
at z=1.579. By relating those systems to their corresponding distribution
functions of being the most massive system in a given survey area, we find that
none of the systems alone is in extreme tension with LCDM. We confront these
results with a GEV analysis of four very massive low redshift clusters: A2163,
A370, RXJ1347-1145 and 1E0657-558, finding no tendency of the high-z systems to
be more extreme than the low-z ones. In addition, we study the extreme
quantiles of single clusters at high-z and present contour plots for fixed
quantiles in the mass vs. survey area plane for four redshift intervals,
finding that, in order to be significantly in conflict with LCDM, cluster
masses would have to be substantially higher than the currently observed ones.
