In NYS, you cannot just terminate their lease. If they have been there for a day longer than 1 year, they must get 60 days' notice; if they have been there for a day longer than two years, the landlord must give 90 days' notice. However, if this locality has opted in for "just cause eviction" (which is statewide), then you cannot terminate the lease without "just cause." All these statutes supercede any written contract between landlord and tenant.
Now, if they stop paying, then you do not have the restrictions above. However, you must give both a 5 and 14 day notice (concurrently) which must be properly served, and it gets more technical from there. Even if you do not mess up any steps in this lengthy process, you are looking at AT LEAST 90 days before you get your asset back. Since it's winter, it could be a great deal longer. Also, it goes without saying that if you make a procedural error (very likely as the process is designed to trip up the unsophisticated landlord, and all tenants have the right to a pro bono attorney), then you have to start the process again from scratch. I would not do a summary eviction proceeding in NY without an attorney. I would start the process immediately. Hopefully they will see you are not screwing around and will leave on their own.
I hope you got a great deal on this property, because tenants in place are a liability, not an asset, in the Empire State.
Now, who loves NY (besides our many "guests from out of town")?