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 $C(t)\equiv Cov(X(1),X(t+1))\mathrm{/}Var(X(1))$. Furthermore, $z(t)={\sum }_{s=1}^{t}X(s)\mathrm{/}t$ 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 $q(z(t))=(\tanh [J(2z(t)-1)+h]+1)\mathrm{/}2,J\ge 0$, 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 $J=J_c(h)$, $C(t)\simeq c+a\cdot \log (t{)}^{-{\alpha }^{\mathrm{\prime }}}$ and $c=0(c>0),{\alpha }^{\mathrm{\prime }}=0.5(1.0)$ for $h=0(h\mathrm{\ne }0)$. The system shows a continuous phase transition for $h=0$ and $C(t)$ behaves as $C(t)\simeq t^{-{\alpha }^{\mathrm{\prime }}}g((1-l)\log t)$ with an universal function $g(x)$ and a length scale $1\mathrm{/}(1-l)$ with respect to $\log t$. $\beta ={\nu }_{\mathrm{\mid }\mathrm{\mid }}\cdot {\alpha }^{\mathrm{\prime }}$ holds with critical exponent $\beta =1\mathrm{/}2$ and ${\nu }_{\mathrm{\mid }\mathrm{\mid }}=1$.