New York Consolidated Laws, Election Law - ELN § 6-162. Primary;  New York City, run-off

1. In the city of New York, when no candidate for the office of mayor, public advocate or comptroller receives forty percent or more of the votes cast by the members of a political party for such office in a city-wide primary election, the board of elections of such city shall conduct a run-off primary election between the two candidates receiving the greatest number of votes for the same office.

2. In any jurisdiction that authorizes a run-off election after a primary election, if one of the two candidates receiving the greatest number of votes for the same office files with the local board of elections a certificate of withdrawal within three days following such primary election, the board shall accept and certify the withdrawal and declare the remaining candidate the winner and no such run-off primary election shall be held.  Such certificate of withdrawal shall be in affidavit or affirmation form as determined by the state board of elections.