Correlation function for generalized P\'olya urns: Finite-size scaling analysis

We describe a universality class of the transitions of a generalized P\'{o}lya urn by studying the asymptotic behavior of the normalized correlation function $C(t)$ using finite-size scaling analysis. $X(1),X(2),\cdots$ are the successive additions of a red (blue) ball [$X(t)=1\,(0)$] at stage $t$ and $C(t)\equiv \mbox{Cov}(X(1),X(t+1))/\mbox{Var}(X(1))$. Furthermore, $z(t)=\sum_{s=1}^{t}X(s)/t$ represents the successive proportions of red balls in an urn to which, at the $t+1$-th stage, a red ball is added, [$X(t+1)=1$], with probability $q(z(t))=(\tanh [J(2z(t)-1)+h]+1)/2,J\ge 0$, and a blue ball is added, [$X(t)=0$], with probability $1-q(z(t))$. A boundary $(J_{c}(h),h)$ exists in the $(J,h)$ plane between a region with one fixed point and another region with two stable fixed points for $q(z)$. $C(t) \sim c+a\cdot t^{l-1}$ with $c=0\,(>0)$ for $J<J_{c}\,(J>J_{c})$, and $l$ is the (larger) value of the slope(s) of $q(z)$ at the stable fixed point(s). On the boundary $J=J_{c}(h)$, $C(t)\simeq c+a\cdot \log(t)^{-\alpha'}$ and $c=0\,(c>0), \alpha'=0.5\,(1.0)$ for $h=0\,(h\neq 0)$. The system shows a continuous phase transition for $h=0$ and $C(t)$ behaves as $C(t)\simeq t^{-\alpha'}g((1-l)\log t)$ with an universal function $g(x)$ and a length scale $1/(1-l)$ with respect to $\log t$. $\beta=\nu_{||}\cdot \alpha'$ holds with critical exponent $\beta=1/2$ and $\nu_{||}=1$.