Continuum limit

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.

See also

• Universality classes

References

• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
• H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I and Vol. II)
• H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online)