Randy, like Page, was a unique individual when it came to recording. What I mean by this is, he would us a lot of unconventional techniques to record his guitar. Sometimes, he would record the same solo, note for note, several times with different mic's, mic placement, cabinets, rooms, etc... and combine them all for one solo track. This is part of his unique solo tone. He also literaly had a castle for a home (if I remember correctly). So, he would set his amps up in a very larg room with stone walls and place several different mic's in differenct places in the room. The room itself was like a natural echo chamber, so each mic would pick up a different tone. He used highly cranked Marshall heads, and like Blues_Man said, the mids would most likely be cranked to the max. There are a few tricks to achieve his tone at home though. First of all, an amp with a nice Marshall type crunch would be nice. A Boss DS-1 distortion pedal can help with this. An eq pedal can also help. I would suggest boosting the mids around 800Hz or so, and cutting the bass a bit. The exact settings will largely depend on the rest of your setup though. To achieve that doubled tone you can use a chorus pedal. Be sure to try it in front of your amp and also in the FX loop. It will sound different in both places. You can also use a delay pedal with the delay time set as low as it will go (20ms or so) and the feedback as low as it will go (so you only get 1 strong repeat), and then set the level so that the delayed signal is equal to the normal signal. This will also give you a "doubled" chorus type sound. Again, it will sound different infront of the amp, and in the FX loop, so try both.

Apart from all that, you'll never sound like Randy, unless you learn to play like Randy. A big part of your tone comes from your hands, and all the little details you wouldn't normally think about.

I hope some of that helps, but if not, there's you a free lesson in music history 101 :D