Critical probability distributions of the order parameter at two loops: I. Ising universality class

Abstract There exists an entire family of universal probability distribution functions (PDFs) of the magnetization mode of the three dimensional Ising model parameterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits="true">lim</mml:mo> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:munder> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> which is the ratio of the system size L to the bulk correlation length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> with both the thermodynamic limit and the critical limit being taken simultaneously at fixed ζ . Recently, the PDFs of the magnetization mode of the three-dimensional Ising model has been computed at one-loop in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>−</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> expansion (2024 arXiv preprint arXiv:2407.12603 ). We show how these PDFs or, equivalently, the rate functions which are their logarithm, can be systematically computed at second order of the perturbative expansion. We compute the whole family of universal rate-functions and show that their agreement with the Monte Carlo data improves significantly at this order when compared to their one-loop counterpart.