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We study the phase transition and the critical properties of a nonlinearP\'{o}lya urn, which is a simple binary stochastic process $X(t)\in\{0,1\},t=1,\cdots$ with a feedback mechanism. Let $f$ be a continuous functionfrom the unit interval to itself, and $z(t)$ be the proportion of the first $t$variables $X(1),\cdots,X(t)$ that take the value 1. $X(t+1)$ takes the value 1with probability $f(z(t))$. When the number of stable fixed points of $f(z)$changes, the system undergoes a non-equilibrium phase transition and the orderparameter is the limit value of the autocorrelation function. When the systemis $Z_{2}$ symmetric, that is, $f(z)=1-f(1-z)$, a continuous phase transitionoccurs, and the autocorrelation function behaves asymptotically as$\ln(t+1)^{-1/2}g(\ln(t+1)/\xi)$, with a suitable definition of the correlationlength $\xi$ and the universal function $g(x)$. We derive $g(x)$ analyticallyusing stochastic differential equations and the expansion about the strength ofstochastic noise. $g(x)$ determines the asymptotic behavior of theautocorrelation function near the critical point and the universality class ofthe phase transition.

Universal function of the non-equilibrium phase transition of nonlinear P\'{o}lya urn