Kiril has discussed this at length in the past and posted many articles.
The short version is that the height of the cut in large part determines how deep the roots will go. YES there are a million other considerations and this is a very, very short version of it all.
But if you mow at 2", the roots are probably around that depth, if you mow at 3" they are about 3" deep. Watering, fertilizer and other practices have a huge impact.
But the gist is that deeper roots are able to access water below the surface, allowing for more drought resistance. Root growth slows greatly in hot weather, but roots are always trying to regenerate.