External monitor has "snow" at 1080p/i when connected to MacBook Air
Solution 1:
Likely to be the adaptor cable if it's cheap - might not be able to handle the HD throughput (although outputting to HDMI, you'd think it'd be able to).
While this sounds counter-intuitive, try moving around the alignment of the second screen as this might ditch the 'snow'.
Also, have you tried the screen with any other HD inputs?
Solution 2:
In my experience this can also be caused by something as simple/stupid as having your HDMI cable rolled up. The EM field put out by the signal traveling through the cable can throw off the integrity of the signal itself... so, unwrap your cable if you havent already :)
Solution 3:
I changed the cable to the monitor buying a £8 ($12) cable from Amazon and it stopped immediately.
The cable prior that I was using was a Belkin one that lasted for 4 years and was 5 times the price, before the snow started and just got worse.