Tu.114 & Chris: I believe you are right, new aircraft tires are simply inflated using nitrogen initially.
tdracer: You're conflating molecular weight and molecular size (kinetic diameter). N2 weighs less than O2, but N2 is (very slightly) larger in size than O2.
However, I agree that the effect on leak loss is likely negligible for all intents and purposes.
For racing, using nitrogen can indeed reduce pressure fluctuations.
If you fill tires with regular air then you're subject to whatever humidity level is in the ambient air at the time. On humid days this can result in a lot of water vapor in the tires.
At race speeds, the tire's internal temperature can rise to more than 100 C, the boiling point of water. This causes the water vapors to expand rapidly and the tire pressure can increase significantly. Hence the use of dry nitrogen instead of regular air.