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I believe it does not say in the book that Arwen's faith is linked to the ring's, nor Elrond's comment on the grace of the Valar leaving her. Book Arwen is not as much a principal character as in the movie.
Back on topic a bit, I don't think that Arwen by choosing to be mortal she chose to live a short life. We know that when King Eleassar lays down to die (as he is allowed to choose his time of death) Arwen begs him not to die and then we see her running to Lorien, as ElentariGreenleaf said, not to be seen ever again. Yet we don't know if she died of "natural causes" or if something (sadness maybe?) killed her.
Perhaps Gurthang is on to something when he says that she would still not die "just because" but rather that the difference would be after her death (caused by some external factor)
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