Having never been a runner, only doing it in the pursuit of other things, I don't have the hunger to run on a daily basis. I do, however have that hunger for martial arts and other things and have empathy for anyone returning to their passions.
I do believe that I needed to temper that with caution, given the stakes at hand. The statistics, the doctors all point to healing not being fully completed until two years. The direction I was given by my surgeon, who has pretty aggressive recovery protocols, is that I wasn't to do any contact sports until one year had passed. He was leery of running until the year had passed, but definitely no martial arts, soccer, etc.
The way I've heard it, the bone needs that time to grow into the prosthesis. I believe most of us have uncemented cups and cemented femoral components, so the annealing that has to happen is unaided bone to the cup and its interstices, bone to glue or in the case of some, straight to the device again. Nutrition might help in the process, but the time table of that healing is not affected by anything we can do. We can help by protecting the process.
I spent long years working out at my different sports. Each of them demanded a lot of effort and I became adept at pushing through pain, based on the idea that improvement only came with overcoming the body's responses, no pain, no gain. I surged through broken ribs (three) dislocated hands and shoulder, broken toes, hamstring pulls and one truly epic shin injury that turned my entire lower leg black; a long and labored list. I don't regret a single one of those, things happen in the course of being active. I have been very aggressive with my body - coupled with my high tolerance for pain (a trait my regular doctor laments), does not lend itself to being patient.
I can understand that for many of us this patience is the most difficult tactic of all that we use in recuperating. We are, though confronted with a simple fact: bone grows at its own pace and even the hardiest of us can't change its nature. Take the long years of your OA suffering, the hejira of the surgery and recovery and the brightness of what you can do in just a comparatively small space of time and patience in that context seems to be your best friend.
If your body feels pain, it is telling you to ease off. In the case of a fully recovered, healthy hippy, I would still take its advice and modulate your activities. I've had my setbacks and recovered, but only because I listened to my body; I still do.
I got back to walking / running on a treadmill at 10 months or so after my second, but it really was mostly walking, mixed in with a minute or so of running. I found it difficult, but not painful before or after. Combining it with any weight lifting with legs did cause be a couple of setbacks so I backed off. So the running / walking was done in fits and starts for the next two months. Again, never been a runner, but did want to at least get functional with it. Soccer looked likely in the spring, and I wanted some comfort with running. I intensified it after a year, but still did walk/run/walk and to tell you the truth, I'll stay with that.
I got a full hour of soccer running in, but not until 16 months on my last hip and 19 on the first. The same with martial arts, didn't get back to punching back workouts until about 11 months and just now started kicking the heavy bag at almost two years. The motion is there, the supporting muscles are weak, but no pain during or after.
This long post is to reinforce what both Pat and John have said, as applied to my recovery. Patience doesn't only apply in the beginning, but is both most difficult and most needed when we feel better and are happy and excited. Keep the true time table in mind and you'll be well served.