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The probability density function (PDF) of some global average quantity playsa fundamental role in critical and highly correlated systems. We explicitlycompute this quantity as a function of the magnetization for the twodimensional XY model in its harmonic approximation. Numerical simulations andperturbative results have shown a Gumbel-like shape of the PDF, in spite of thefact that the average magnetization is not an extreme variable. Our analyticalresult allows to test both perturbative analytical expansions and alsonumerical computations performed previously. Perfect agreement is found for thefirst moments of the PDF. Also for large volume and in the high temperaturelimit the distribution becomes Gaussian, as it should be. In the lowtemperature regime its numerical evaluation is compatible with a Gumbeldistribution.

Analytical computation of the magnetization probability density function for the harmonic 2D XY model