Paul is one of Lucy's lovers whom she first meets at a party. Paul has a reputation as a perv, at least according to Peggy. Despite this warning, Lucy is super attracted to the guy and begins a sexual relationship with him.
It's a pretty hot and heavy relationship, too. Lucy confides in Mariah:
I told [Mariah] what everything felt like, how surprised I was to be thrilled by the violence of it (for sometimes it was that, violent), what an adventure this part of my life had become. (4.30)
Lucy's relationship with Paul is part of a bigger pattern of her relationships with men in the novel. As in her previous relationship with Hugh, Lucy doesn't need to be head over heels in love to enjoy sex. She tells us:
The question of being in love was not one I wanted to settle then; what I wanted was to be alone in a room with [Paul] and naked. (4.19)
At least she's got her priorities straight.
But it's not like Lucy is playing some game of keeping herself emotionally removed from Paul because she's afraid he doesn't care for her. After he tells her that he loves her, she says that she "kissed him doubly hard, and instantly I knew it was a mistake, for he mistook my enthusiasm for his love returned" (4.34).
What a little heartbreaker that Lucy is.