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

• Received 17 November 2020
• Revised 11 June 2021
• Accepted 11 June 2021

DOI:https://doi.org/10.1103/PhysRevE.104.014109