It depends on where the guards are
If the guards are in the city, then that's a big prison. You can expect to see the same kinds of prison riots that humans have observed in all kinds of actively oppressed populations going back to antiquity. Expect this kind of situation if the arcologies are actively trying to punish the prisoners. Oppression rarely works out well.
Given the scale of the penal institutions, it will be very difficult for the prison's security apparatus to enforce uniform control. Expect to see the emergence of no-go zones where the prison authority cannot exercise full control.
Personally, I think this is the least likely option since it costs so much to hire and equip guards. Also, if this is a true prison population then the arcologies are on the hook to provide food and medical care for the inmates. If you're populating New York purely with inmates that have to be supported, your arcologies are either wealthy beyond reckoning or this program won't last long.
If the guards are around the arcologies to prevent the prisons from getting back in, then it's possible that we'll get an Australia style penal colony. When given a chance to start new lives, the former prisoners started businesses, married, had children and built a civilization in the middle of nowhere.
I think this is the more likely as it's cheaper to just eject people you don't like from the arcologies and make them fend for themselves. Given how well humans are at surviving, the inmates/undesirables will find a way. If you did it right, these ex-arcology people would be a useful source of trade and "off-shore" labor for the arcologies.