We present the recent merger history of massive galaxies in a spectroscopically-confirmed proto-cluster at z=1.62. Using HST WFC3 near-infrared imaging from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS), we select cluster galaxies and z ~ 1.6 field galaxies with M_star >= 3 x 10^10 M_sun, and determine the frequency of double nuclei or close companions with projected separations less than 20 kpc co-moving and stellar mass ratios between 1:1 and roughly 10:1. We find that four out of five spectroscopically-confirmed massive proto-cluster galaxies have double nuclei, and 42 +13/-25 % of all M_star >= 3 x 10^10 M_sun cluster candidates are either in close pair systems or have double nuclei. In contrast, only 4.5 +/- 2.6% of the field galaxies are in close pair/double nuclei systems. The implied merger rate per massive galaxy in the proto-cluster is 3-10 times higher than the merger rate of massive field galaxies at z ~ 1.6, depending upon the assumed mass ratios. Close pairs in the cluster have minor merger stellar mass ratios (M_primary:M_satellite ~ 6:1), while the field pairs are typically major mergers with stellar mass ratios between 1:1 and 4:1. At least half of the cluster mergers are dissipationless, as indicated by their red colors and low 24 micron fluxes. Two of the double-nucleated cluster members have X-ray detected AGN with L_x > 10^43 erg/s, and are strong candidates for dual or offset super-massive black holes. We conclude that the massive z = 1.62 proto-cluster galaxies are undergoing accelerated assembly relative to the field population, and discuss the implications for galaxy evolution in proto-cluster environments.
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