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

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=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≥0, and a blue ball is added [X(t+1)=0] with probability 1-q[z(t)]. A boundary [J(c)(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'·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)≃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)lnt] with a universal function g(x) and a length scale 1/(1-l) with respect to lnt. β=ν(||)·α' holds with β=1/2 and ν(||)=1.