Statistical mechanics and quantum fields

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At carefully adjusted temperature and pressure liquid and steam phases of water have the same density, and water randomly fluctuates on all length scales between the two phases. Systems that fluctuate on all scales are said to be at a "critical point". Motivated by a conjecture that many different systems can be united under a small list of such laws, a primary goal of statistical mechanics is to calculate probability laws of scale invariant fluctuations. The Nobel prize winning work of Ken Wilson in 1982 began a program called the "Renormalisation Group" aimed at the creation of a mathematical theory of these scale invariant probability laws. This proposal will advance Wilson's program. Since critical points unite many systems it is reasonable to start with the simplest representative cases, one of which is self-avoiding walk. A self-avoiding walk is scale invariant in the sense that individual steps of a very long self-avoiding walk seen from far away are invisible and all that can be seen is a continuous curve that has "wiggles" on all scales. Thus we are initially focused on the question of how the end-to-end distance of a self-avoiding walk grows as a function of the number of steps.