Abstract
We give a survey and unified treatment of functional integral 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 introduction to fermionic integrals. The fermionic integrals are in terms of anticommuting Grassmann variables, which can be conveniently interpreted as differential forms.
Highlights
The use of random walk representations for functional integrals in mathematical physics has a long history going back to Symanzik [25], who showed how such representations can be used to study quantum field theories
Our goal in this paper is to provide an introductory survey of functional integral representations for some problems connected with self-avoiding walks, with both strict and weak self-avoidance
We derive a new representation for the strictly self-avoiding walk. These representations have proved useful recently in the analysis of various problems concerning 4-dimensional selfavoiding walks, by providing a setting in which renormalization group methods can be applied. This has allowed for a proof of |x|−2 decay of the critical Green function and existence of a logarithmic correction to the end-to-end distance for weakly self-avoiding walk on a 4-dimensional hierarchical lattice [3, 6, 7]
Summary
The use of random walk representations for functional integrals in mathematical physics has a long history going back to Symanzik [25], who showed how such representations can be used to study quantum field theories. The bosonic representations will be the most familiar to probabilists, as they are in terms of ordinary Gaussian integrals They represent simple random walks, and systems of self-avoiding and mutually-avoiding walks and loops. The mixed bosonic-fermionic representations eliminate the loops, leaving only the self-avoiding walk They involve Gaussian integrals with anticommuting Grassmann variables. These are purely bosonic representations, without anticommuting fermionic variables. An appreciation of this fact is not necessary to understand the representations, in Section 6 we briefly discuss this important connection
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