What's the law on rent increase when a tenent leaves?

[landy]

The only lawful way to evict a tenant is for the landlord to obtain a court order signed by the circuit court or magistrate judge. This is obtained in a lawsuit called a "forcible entry and detainer" action. After giving a three-day notice, a landlord can secure a court order to have a tenant evicted if either:

The tenant is in unlawful possession of the landlord's property (by remaining on the property after the expiration of a rental agreement or failing to pay rent for more than three days after it is due);

The tenant substantially damages the premises; or

The tenant does or fails to do something which, under the terms of the lease, is identical to cancellation.

A tenant must be given three (3) days notice to vacate before a forcible entry and detainer action can be commenced by a landlord. the tenant refuses to move after three days, the landlord can then file a lawsuit for eviction.

[elklaw]

Typically, eviction is a result of a judgment rendered by a court that has jurisdiction to issue writs of eviction.