Your online rating, results, ladder points and cloud save are deleted from the server immediately, and the game starts again as a new guest.
To ask us to delete your data, use the developer contact email on the Google Play listing with the subject "Delete my Gambit Arena account". Include the player id shown in Settings → About or the Google account you signed in with. We delete the data within 30 days.
All of it: rating, game results, ladder and arena records, and the cloud copy of your progress. Nothing is kept afterwards, apart from server logs that expire within 30 days.