Yanagita shows up in one of the weirdest parts of the movie. As Ethan Coen noted, " It was certainly our intention while writing this sequence that it should be a digression."
While in Minneapolis, trying to crack the case, Marge meets up with Mike, an old friend from high school. Mike heard about her while watching TV news about the murders up in Brainerd. They eat dinner at the Radisson, where Mike ineptly tries to seduce her. He tells Marge he's working as an engineer at Honeywell, then claims that he's felt so lonely since his wife, Linda (who Marge also knew in high school) died:
MIKE: Yah, well, I, uh... it's not that it didn't work out... Linda had leukemia. She passed away.
However, Marge later discovers that Mike made all this up; he was never married to Linda, who's still very much alive but had to move away because Mike was stalking her. Mike apparently has psychiatric problems and is now living with his parents, according to a friend.
Why is this in the movie? Maybe the Coens just wanted to mess with people and have them debate. The bros themselves said it was just to give us a look at Marge's life outside of her job. Some critics believe that Marge's realization that she was tricked made her think that maybe Jerry was lying to her as well, and that's why she went to see him a second time.