<p>The competitive applicant with the reputed, respected advisor who is an alumnus of the school maybe has a better chance than the same, equally competitive applicant in an alternate universe where the same equally reputed, respected advisor got his PhD from some other institution.</p>
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If the current faculty don’t already know, I expect a professor writing a letter to an alma mater would include something to the effect of “As an alumnus of this program, I know [student] is/will…”</p>