# Universality and the continuum limit

# Universality and the continuum limit

This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality using the classical example of the central limit theorem in probability theory. It then discusses the properties of the random walk on a lattice, where universality is directly related to the continuum limit. In both examples, the chapter is interested in the collective properties of an infinite number of random variables in a situation where the probability of large deviations with respect to the mean value decreases fast enough. They differ in the sense that a random walk is based on a spatial structure that does not necessarily exist in the case of the central limit theorem. From the study of these first examples emerges the importance of Gaussian distributions, and this justifies the technical considerations of Chapter 2. The chapter introduces some transformations, acting on distributions, which decrease the number of random variables. It shows that Gaussian distributions are attractive fixed points for these transformations. This will provides the first, extremely simple, applications of the renormalization group (RG) ideas and allows the establishment of corresponding terminology. Finally, in this context of the random walk, a path integral representation is associated with the existence of a continuum limit. Exercises are provided at the end of the chapter.

*Keywords:*
universality, continuum limit, central limit theorem, random walk, Brownian motion, path integrals

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