something i'm confused about

before all the comments start saying the same thing: i know this isn’t the focus point of the show. i know it isn’t some crime thriller drama. that being said; how did the police manage to figure out that the perpetrator was jamie so quickly? there were about 12 hours between the murder and jamie's arrest so how could they have linked it to jamie with just the cctv footage and trainers? you couldn’t see his face on the cctv