We review recent progress in limit laws for the one-dimensional asymmetric simple exclusion process (ASEP) on the integer lattice. The limit laws are expressed in terms of a certain Painlevé II function. Furthermore, we take this opportunity to give a brief survey of the appearance of Painlevé functions in statistical physics.
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Craig A. Tracy, Harold Widom, Painlevé Functions in Statistical Physics. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 361–374DOI 10.2977/PRIMS/38