Congrats!! I'm not in Epi, but seems like Epi departments in general are really moving things along.
Per u/FrozenFreshMeat's post, did you get an automated email from the university/department, or a direct email from a potential advisor?
I don't think it's "desperate" to email them to say thanks and mention that you submitted your app, but honestly, I wouldn't expect much/anything to be gained from sending that email. It's just being polite.