Why does she stare at me but never talk to me?

This girl always turns her head around to look at me. She'll even go out of her way to walk past me. We just see each other around and have never had a real conversation. I asked her out to lunch about two weeks after getting her name and got rejected. But after the rejection she still does the same thing. We'll make eye contact in places like the library but neither of us will say hi to each other first. I personally don't say hi first so I don't seem desperate. The funny thing is a few minutes later she'll make it so obvious she's going out of her way to walk right in front of me. Some tables in the library are kinda close to each other so you have to squeeze through to walk through some of them, which is why the vast majority of people walk around the tables. Out of all the table she chose to walk through, she chose to squeeze in between my table right in front of me. She did this without making any eye contact, which is strange considering the fact that I would catch her staring at me across the room every few minutes. What is she doing?! She rejected lunch with me but makes it very obvious that she goes out of her way to get my attention. She never even initiates a conversation with me. And what should I do?
Why does she stare at me but never talk to me?
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