I went to Best Buy and they gave me a new one, but it had the same problem. I left my phone home one day, took out the sim card and left the wifi on with sync off (I believe). It only lost about 3% in 15 hours, leading me to believe that while it shows wifi is doing something, it isnt doing much to effect battery life. Regardless, the down arrow shouldn't always be on like that, and there's still some tweaking Samsung can do to get better battery life (cell standby issues). Battery life also improves if you disable S Voice btw.