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We investigate the density decay in the pair-annihilation process A+A->0 inthe case when the particles perform anomalous diffusion on a cubic lattice. Theanomalous diffusion is realized via L\'evy flights, which are characterized bylong-range jumps and lead to superdiffusive behavior. As a consequence, thecritical dimension depends continuously on the control parameter of the L\'evyflight distribution. This instance is used to study the system close to thecritical dimension by means of the nonperturbative renormalization grouptheory. Close to the critical dimension, the assumption of well-stirredreactants is violated by anticorrelations between the particles, and the law ofmass action breaks down. The breakdown of the law of mass action is known to becaused by long-range fluctuations. We identify three interrelated consequencesof these fluctuations. First, despite being a nonuniversal quantity and thusdepending on the microscopic details, the renormalized reaction rate can beapproximated by a universal law close to the critical dimension. The emergenceof universality relies on the fact that long-range fluctuations suppress theinfluence of the underlying microscopic details. Second, as criticality isapproached, the macroscopic reaction rate decreases such that the law of massaction loses its significance. And third, additional nonanalytic power lawcorrections complement the analytic law of mass action term. An increasingnumber of those corrections accumulate and give an essential contribution asthe critical dimension is approached. We test our findings for twoimplementations of L\'evy flights that differ in the way they cross over to thenormal diffusion.

Fluctuation Effects in the Pair-Annihilation Process with L\'evy Dynamics