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

Shintaro Mori and Masato Hisakado
Phys. Rev. E 92, 052112 – Published 10 November 2015

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,J0, and a blue ball is added [X(t+1)=0] with probability 1q[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·tl1 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(h0). The system shows a continuous phase transition for h=0 and C(t) behaves as C(t)(lnt)αg[(1l)lnt] with a universal function g(x) and a length scale 1/(1l) with respect to lnt. β=ν||·α holds with β=1/2 and ν||=1.

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  • Received 10 June 2015
  • Revised 6 October 2015

DOI:https://doi.org/10.1103/PhysRevE.92.052112

©2015 American Physical Society

Authors & Affiliations

Shintaro Mori*

  • Department of Physics, Kitasato University, Kitasato 1-15-1, Sagamihara, Kanagawa 252-0373, Japan

Masato Hisakado

  • Financial Services Agency, Kasumigaseki 3-2-1, Chiyoda-ku, Tokyo 100-8967, Japan

  • *mori@sci.kitasato-u.ac.jp

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Vol. 92, Iss. 5 — November 2015

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