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Symmetries play a conspicuous role in the large-scale behavior of criticalsystems. While in equilibrium they allow to classify asymptotics into differentuniversality classes, out of equilibrium they can emerge, some timesunexpectedly, as collective properties which are not explicit in the 'bare'interactions. Here we elucidate the emergence of an up-down symmetry in theasymptotic behavior of the stochastic scalar Burgers equation in one and twodimensions, manifested by the occurrence of Gaussian fluctuations for thephysical field. This robustness of Gaussian behavior contradicts naiveexpectations, due to the detailed relation ---including the same set ofsymmetries--- between Burgers equation and the Kardar-Parisi-Zhang equation,which paradigmatically displays non-Gaussian fluctuations described byTracy-Widom distributions. We reach our conclusions via a dynamicrenormalization group study of the field statistics, confirmed by directevaluation of the field probability distribution function from numericalsimulations of the dynamical equation. All odd-order cumulants cancel exactly,while the excess kurtosis vanishes for large systems, indeed consistent withGaussian behavior.

Gaussian statistics as an emergent symmetry of the stochastic Burgers equation