Many stochastic algorithms reach the desired distribution of random variables only asympoticaly.  When sampling is ergodic the  properties extracted by the simulations are independent of the starting configuration. To make sure your simulation has reached the asymptotic limit you have to  run a number of moves without accruing estimates. 200 moves seems too small to me. I typically run about 1000000 moves before I begin accumulating estimates. You will be able to tell if sampling and if sufficient steps to reach the asymptotic limit by looking the reproducibility of your results. Run you simulation again with a different starting configuration

Ok, I have a code and I want to make a simulation having 5000 repetitions and then I calculate the average MSE of the estimators. But what do you mean "moves before simulation". Should I run the code 1000000 times and then start the 5000 replications?

I did not get the point, sorry.

I am assuming you are attempting to convert uniform random distributions into distributions you desire (e.g. the Boltzmann distribution).  You simply need to draw a number of these random numbers until the estimators reach an equilibrium state. (Graph your estimator as a function of the number of moves). You will see what I mean.