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We study the red sequence in a cluster of galaxies at z=1.62 and follow itsevolution over the intervening 9.5 Gyr to the present day. Using deep YJKsimaging with the HAWK-I instrument on the VLT we identify a tight red sequenceand construct its rest-frame i-band luminosity function (LF). There is a markeddeficit of faint red galaxies in the cluster that causes a turnover in the LF.We compare the red sequence LF to that for clusters at z<0.8 correcting theluminosities for passive evolution. The shape of the cluster red sequence LFdoes not evolve between z=1.62 and z=0.6 but at z<0.6 the faint populationbuilds up significantly. Meanwhile, between z=1.62 to 0.6 the inferred totallight on the red sequence grows by a factor of about 2 and the bright end ofthe LF becomes more populated. We construct a simple model for red sequenceevolution that grows the red sequence in total luminosity and matches theconstant LF shape at z>0.6. In this model the cluster accretes blue galaxiesfrom the field that are then quenched and subsequently allowed to merge. Wefind that 3--4 mergers among cluster galaxies during the 4 Gyr between z=1.62and z=0.6 matches the observed luminosity function evolution between the tworedshifts. The inferred merger rate is consistent with other studies of thiscluster. Our result supports the picture that galaxy merging during the majorgrowth phase of massive clusters is an important process in shaping the redsequence population at all luminosities.

A Tale of Dwarfs and Giants: Using a z=1.62 Cluster to Understand How the Red Sequence Grew Over The Last 9.5 Billion Years