Perturbative expansions in many physical systems yield "only" asymptotic series which are not even Borel resummable. Interestingly, the corresponding ambiguities point to nonperturbative physics. We numerically verify this renormalon mechanism for the first time in two-dimensional sigma models, that, like four-dimensional gauge theories, are asymptotically free and generate a strong scale through dimensional transmutation. We perturbatively expand the energy through a numerical version of stochastic quantization. In contrast to the first energy coefficients, the high-order coefficients are independent on the rank of the model. Technically, they require a sophisticated analysis of finite volume effects and the continuum limit of the discretized model. Although the individual coefficients do not grow factorially (yet), but rather decrease strongly, the ratios of consecutive coefficients clearly obey the renormalon asymptotics. [ABSTRACT FROM AUTHOR]