Because of the Feynman path integral, QFT can be made into a statistical field theory. In rigourous relativistic field theories, this is formalized by the Osterwalder-Schrader conditions. At any rate, there are well established links between quantum field theory and statistical physics. A famous equation in statistical physics is the KPZ equation, which appears to involve the product of distributions and so isn't obviously well defined, which of course hasn't stopped physicists from finding it meaningful. One of this year's Fields Medals was given to Martin Hairer, apparently for being able to make rigourous sense of the KPZ equation using Terry Lyons's "rough paths" theory. In this abstract for a talk by Lyons http://www.oxford-man.ox.ac.uk/events/what-can-rough-paths-do-for-you, it is mentioned that rough paths theory is related via Hairer's work to "John Cardy's work on constructive conformal field theory". Is there any simple introduction to what this means?