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

Publisher Website
Google Scholar
Abstract
We describe a universality class for 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),... are the successive additions of a red (blue) ball [X(t)=1(0)] at stage t and C(t)≡Cov[X(1),X(t+1)]/Var[X(1)]. Furthermore, z(t)=∑s=1tX(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≥0, and a blue ball is added [X(t+1)=0] with probability 1−q[z(t)]. A boundary [Jc(h),h] exists in the (J,h) plane between a region with one stable fixed point and another region with two stable fixed points for q(z). C(t)∼c+c′·tl−1 with c=0(>0) for J<Jc(J>Jc), and l is the (larger) value of the slope(s) of q(z) at the stable fixed point(s). On the boundary J=Jc(h),C(t)≃c+c′·(lnt)−α′ and c=0(c>0),α′=1/2(1) for h=0(h≠0). The system shows a continuous phase transition for h=0 and C(t) behaves as C(t)≃(lnt)−α′g[(1−l)
Funding Information
  • Japan Society for the Promotion of Science (25610109)

This publication has 28 references indexed in Scilit: