Keep in mind... this is a rumor, and we are on a student doctor message board - not at a Columbia ADCOM meeting.
I'm a second year at P&S, as well as a tour guide, so I work closely with the admissions department. The LAST thing the ADCOM wants is for people to write letters and letters and letters into Dean Frantz. It's helpful yes, in gauging who has a strong interest in the school -- as the ADCOM naturally would like to select a student body who really loves Columbia... after all, it makes for a happier student body in general. It is NOT helpful if every Tom, Dick and Harry writes a letter. So in this sense, writing a letter to Dean Frantz or anyone else does not carry more weight than it would at any other school.
But as a student at P&S, who knows several individuals in different classes who did NOT write any letters to Dean Frantz and were accepted, I'm telling you to not believe the here-say. If you love the school, sure- let them know. Just like you would anywhere else.
People seem to have a lot of beef with Columbia on this board. Before you agree with it, I suggest looking at whom it's coming from... Usually it's people who "have heard" this or that.