Sommaire:

Random walks with interactions, in particular self-avoiding walks, are ubiquitous. They occur for example in probability theory and statistical physics where they furnish, just like ferromagnetic systems, an important example of the theory of critical phenomena. It can be shown that the Green's function of a large class of weakly self-avoiding (self-repelling) walks in a regular lattice, including those with long range jumps (Levy walks), can be expressed as a two point correlation function in a special kind of supersymmetric field theory. This opens the possibility of applying field theoretic methods, in particular those of the renormalization group, in the study of the scaling limit of such walks. In the case of weakly self-repelling walks with long range jumps a new parameter $\epsilon$ appears such that for $\epsilon$ > 0 one is below the critical dimension of the model but remaining in fixed spatial dimension. In this case, unlike that of Wilson and Fisher for ferromagnetic systems, the $\epsilon$ expansion with remainder can be mathematically controlled in the scaling limit with help of recently developed rigorous renormalization group methods. There is an underlying non-Gaussian fixed point. The exponent $\eta$ vanishes, whereas the exponent $\nu$, and thus the Haussdorf dimension of such walks, develops an anomaly. In this talk I will review this subject including recent results.

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