Functional integral representations for self-avoiding walk ∗

We give a survey and unified treatment of functional inte- gral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self- avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an in- troduction to fermionic integrals. The fermionic integrals are in terms of anticommutingGrassmann variables,which can be convenientlyinterpreted as differential forms.