In this work we present an application of general extreme value (GEV) statistics 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 M200 m= (2.16 ± 0.32) × 1015 M⊙ at a redshift of z= 0.87, SPT-CL J2106−5844 with a mass of M200 m= (1.27 ± 0.21) × 1015 M⊙ at z= 1.132 and two clusters found by the XMM–Newton Distant Cluster Project survey: XMMU J2235.32557 with a mass of located at a redshift z= 1.4 and XMMU J0044.0−2033 having a mass in the range of at z= 1.579. By relating those systems to their corresponding distribution functions of being the most massive system in a given survey area and redshift interval, we find that none of the systems alone is in tension with Λ cold dark matter (ΛCDM). We confront these results with a GEV analysis of four very massive low-redshift clusters: A2163, A370, RXJ 1347−1145 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 versus survey area plane for four redshift intervals, finding that, in order to be significantly in conflict with ΛCDM, cluster masses would have to be substantially higher than the currently observed ones.