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We theoretically study divergent fluctuations of dynamical events atnon-ergodic transitions. We first focus on the finding that a non-ergodictransition can be described as a saddle connection bifurcation of an orderparameter for a time correlation function. Then, following the basic idea ofGinzburg-Landau theory for critical phenomena, we construct a phenomenologicalframework with which we can determine the critical statistical properties atsaddle connection bifurcation points. Employing this framework, we analyze amodel by considering the fluctuations of an instanton in space-timeconfigurations of the order parameter. We then obtain the exponentscharacterizing the divergences of the length scale, the time scale and theamplitude of the fluctuations of the order parameter at the saddle connectionbifurcation. The results are to be compared with those of previous studies ofthe four-point dynamic susceptibility at non-ergodic transitions in glassysystems.Comment: 5 pages. Minor revision. This version will appear in Europhys. Let

A theory for critically divergent fluctuations of dynamical events at non-ergodic transitions