Do I retain my rank and stats across platforms in Battlefield 3?
No, your stats do not carry over platforms.
I can't find an official word from DICE, but I know they don't from experience.
I wish they did, I just built a PC and I'm rank 45 on the 360 :(
@Emerica is correct: No they do not carry over platforms. If you visit BF3's Battlelog and after you login, you will notice next to the player's origin name, which platform he/she is playing on with that account: PS3/XBOX360/PC. You can link origin accounts but you cannot carry over earned rank from one platform to the other.