Scaling limit

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An animated example of a Brownian motion-like random walk on a torus. In the scaling limit, random walk approaches the Wiener process according to Donsker's theorem.

In physics or mathematics, the scaling limit is a term applied to the behaviour of a lattice model in the limit of the lattice spacing going to zero. A lattice model which approximates a continuum quantum field theory in the limit as the lattice spacing goes to zero corresponds to finding a second order phase transition of the model. This is the scaling limit of the model. It is often useful to use lattice models to approximate real-world processes, such as Brownian motion. Indeed, according to Donsker's theorem, the discrete random walk would, in the scaling limit, approach the true Brownian motion.

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