They'll either call you, or mail you, yes. Eventually.

Wait a week or two, then call them, and ask for a status update, and if they need anything else from you. Eventually, they'll say something like 'we're ready to call your references, but we're backed up. Have one of your references call in and give this file number.'

Get your ref to do so. It's literally a two minute call, including ringing, voice menu navigation, and giving them the reference number. They'll be asked 'Is PatsWin crazy? Are you afraid of him having guns? If he does go crazy, will you call the cops?' and that's that.