# Binder cumulant method for non-Gaussian distributions

In the Ising model, we know that the order parameter $$m$$ has a Gaussian distribution for temperatures below the critical point. Measuring the exact point where this phase transition takes place was always done with the help of the Binder cumulant in my classes, see e.g. this.

I'm currently working on a problem where the order parameter undergoes a second-order phase transition and follows a Tracy-Widom distribution below the critical point.

My question is if I can also use the Binder cumulant to estimate the critical point in this problem or if it only works for Gaussian distributions. Because the way that I thought about it is, that the cumulant basically measures how much my distribution looks like a Gaussian one.

I would really appreciate any thoughts on this, maybe I did not correctly think about the concept of the Binder cumulant.

The Binder cumulant (lets call it $$\mathcal{B}$$) is used to estimate how Gaussian is your distribution. What is worrying about the Binder cumulant is, that in many cases the wrong version is used. Sinde Binder defined the cumulant in the case of the Ising model, where there is a zero-mean distribution, there is no difference between using the forms of the central moment or just the Order Parameters. Let's assume $$s$$ is a general OP (not necessarily spin)

$$\mathcal{B}(s) = 1 - \frac{\langle s^4 \rangle_m}{3 (\langle s^2 \rangle_m)^2}$$.

where that little $$m$$ is crucial. I found that in many papers the $$m$$ is omitted, and $$s = \frac{1}{N} \sum^N_i s_i$$, which doesn't make sense if the distribution is not centered around zero, as it will not give more information than the variance.

So assuming, that we are using the proper form: $$\langle s \rangle^n_m = \langle (s-\langle s \rangle)^n \rangle$$, the binder cumulant ought to signal a deviation from the Gaussian distribution.

How does a first-order transition behave? It will produce a double Gaussian distribution, and when you do a finite volume scaling analysis the peaks should be visible and separate even more. The two peaks are related to the coexistence of the two phases, and in numerical simulations often show up as a tunneling between two states.

In the case of a second-order order phase-transition you will have a Gaussian distribution deep in the phases and a distorted, flat Gaussian at the phase-transition. At the infinite volume limit, it should approach a uniform distribution.

The higher-than-order transitions and crossovers are a bit more tricky because even the susceptibilities (variance) will not necessarily show any scaling behavior, leaving you with simple Gaussian distributions. (not to mention, that if you are not in the continuum technically you have only crossovers....).

Let's see the case of general functions, for example, the one you are interested in. It is not a Gaussian distribution, but it may have some parts which are close to it. I think that in the case of Gaussian like-long tailed distributions, one can also use Binder's argument above, but you have to restrict your generalized Gaussian fit (with varying exponent) to the top of your curve, just "cut your tail". If you use a large portion of your curve and have a nice fit to it, then you could argue for the usage of the Binder cumulant. But it also depends on the fact that what results would you get in the case of a first-order distribution, two peaks, or a general function with a different form?