I am a physics student working on the field of phase transition. At the critical point of 2rd order phase transition, a lot of correlation functions have power law behavior. For instance, let's consider the 2D Ising model, then the two point correlation function decays as: $$ f(r) = a r^{-X_0}(1+b r^{-X_1}+ ...) $$ in the limit of r goes to infinity. Our goal is get the value of $X_0$, which is the so-called critical exponent, other terms are called corrections.
In the Monte Carlo simulation, we can obtain some data by measuring f(r) (with statistical error) at some r, usually, we take r=2,4,8, 16,...,$r_{max}$.
Then what we usually do is put these data {$(r_i, f(r_i), error(r_i) )$} into a program and a fitting ansatz of the form:
$$F(x) = a r^{-X_0}(1+b r^{-X_1}+...) $$ where $b$ and "..." are the so-called subleading terms because they decays faster than the term $a r^{-X_0}$. What the program does is a find some parameter $a, b, X_0, X_1$ which minimize the following cost function: $$ \sum_{i} \frac{(f(r_i)-F(r_i) )^2}{error(ri)^2} $$
Obviously, the $error(r_i)$ can affect the resulting $X_0$. My question is how should we choose the value for $error(i)$ to get the most reasonable $X_0$? (In Monte Carlo, when we have more samples for $f(r_i)$, then the $error(r_i)$ is smaller, so by choose $error(r_i)$, I mean how many samples we should take for each $r_i$.) For example, if we have an extreme small error for small $r_i=2$, then it contributes a lot to the cost function, which is not rational since we only cares about large distance behavior.