Universal function of the nonequilibrium phase transition of a nonlinear Pólya urn

We study the phase transition and the critical properties of a nonlinear Pólya urn, which is a simple binary stochastic process X(t)∈{0,1},t=1,⋯, with a feedback mechanism. Let f be a continuous function from the unit interval to itself, and z(t) be the proportion of the first t variables X(1),⋯,X(t) that take the value 1. X(t+1) takes the value 1 with probability f[z(t)]. When the number of stable fixed points of f(z) changes, the system undergoes a nonequilibrium phase transition and the order parameter is the limit value of the autocorrelation function. When the system is Z_{2} symmetric, that is, f(z)=1-f(1-z), a continuous phase transition occurs, and the autocorrelation function behaves asymptotically as ln(t+1)^{-1/2}g[ln(t+1)/ξ], with a suitable definition of the correlation length ξ and the universal function g(x). We derive g(x) analytically using stochastic differential equations and the expansion about the strength of stochastic noise. g(x) determines the asymptotic behavior of the autocorrelation function near the critical point and the universality class of the phase transition.