Dear XXX,
Thank you for your interest in XXX University's programs in XXX. You can find more information about our PhD program by clicking XXX. Admissions are determined by a committee that meets each January. Professors agree to be supervisors usually after students arrive on campus. I am constantly looking for new PhD students. I am copying this message to XXX, who can answer any additional questions that you may have. XXX University is a clear leader in XXX. Your credentials indicate that you may be a very good fit with one of our programs. We look forward to receiving your application. I strongly encourage you to apply! Attached is a follow-up paper to the paper you cite, along with an Excel file to perform the calculations.
Thanks for your inquiry. Your background looks good to me. I do accept new students next fall and you are welcome to apply. I highly recommend you to submit your application as soon as possible for full consideration.
BTW, could you tell me your GRE scores (percentiles)?
Dear XXX,
Thank you for your interest in our department. You have accumulated an impressive record in your pursuits so far. I hope you will apply and I will keep an eye out for your application. I should note, however, that it is likely to be a very competitive year for our program, and I have a limited capacity to take on new students this year. That does not mean that you won't find a suitable home in this department, only that I am unlikely to be the primary advisor at least at first. We all work pretty cooperatively here though, so by all means pursue other leads you have and we will see how things go.