All of the Ph.D. programs in the U.S. are extremely competitive. The main reason for this is the limited amount of funding available to prospective students. Another reason is that most advisors are unwilling to take on more than 2 or 3 students at a time.
As you suggest, most applicants will have a greater chance of success if they can match their research interests with those of a professor at a particular university. You usually have to contact potential advisors via email, or even better, in person at SVP or similar meetings to see if they are interested in sponsoring your Ph.D. application. You might also consider scheduling a tour of the university so that you can meet all of the faculty in the department.
When I applied for grad school I submitted applications to 6 different universites (3 Master's and 3 Ph.D. programs) and was only accepted by one (Master's program at SDSM&T). Some of the other Master's students here have been trying for a couple of years to get into other doctoral programs without any success.
I guess the main point I'm trying to make here is that it is very hard to get into Ph.D. programs but you will have better success if the faculty know your face (and work). Persistence pays too.
Good luck!!