I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov chain before convergence.
I want to know how T scales with N. Specifically I'd like to know whether T=O(poly(N)) or T=O(exp(N)), anything more specific would be a bonus but is not required. To determine this I have run the program for many values of N and got the corresponding values of T. But how do I determine the complexity from this data? How do I distinguish whether between a high order polynomial and a slow exponential? To what degree is it actually possible to do this?
Note: I have also posted this question in an alternative form at Polynomial and exponential regression