Perturbative PDF of the total magnetization of the 4D Ising model

We compute, at one loop in perturbation theory, the probability density function of the total magnetization $M$ of the Ising model on the 4-torus and the 4-sphere. We develop a single perturbative expansion that is valid in the symmetric phase as well as the broken symmetry phase, provided that the correlation length is large compared to the system size $L$. We find that, at the critical point, for large system size in lattice units, the PDF approaches $p(M)\sim \exp (-f(L)M^4)$. Consequently, the critical value of the Binder cumulant of the total magnetization is $U = 1 - \frac{4,\Gamma(5/4)^2}{3,\Gamma(3/4)^2}$. We validate our results by comparison with Monte Carlo simulation.