Correlation function for generalized Pólya urns: Finite-size scaling analysis

Abstract

We describe a universality class of the transitions of a generalized Pólya 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 $J=J_c(h)$, $C(t)\simeq c+a\cdot \log (t{)}^{-{\alpha }^{\prime }}$ and $c=0(c>0),{\alpha }^{\prime }=0.5(1.0)$ for $h=0(h\ne 0)$. The system shows a continuous phase transition for $h=0$ and $C(t)$ behaves as $C(t)\simeq t^{-{\alpha }^{\prime }}g((1-l)\log t)$ with an universal function $g(x)$ and a length scale $1/(1-l)$ with respect to $\log t$. $\beta ={\nu }_{||}\cdot {\alpha }^{\prime }$ holds with critical exponent $\beta =1/2$ and ${\nu }_{||}=1$.

26 pages, 8 figures

References in corpus (8)

• Statistical physics of social dynamics
• A survey of random processes with reinforcement
• Is the Voter Model a model for voters?
• Controlling Contagion Processes in Time-Varying Networks
• Correlated Binomial Models and Correlation Structures
• Threshold learning dynamics in social networks
• Information cascade, Kirman's ant colony model, and kinetic Ising model
• Finite-size scaling analysis of binary stochastic processes and universality classes of information cascade phase transition

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