Year: 2015
Type: article
Abstract: We describe a universality class for the transitions of a generalized P\'olya 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 $[X(t)=1\phantom{\rule{0.16em}{0ex}}(0)]$ at stage $t$ and $C(t)\ensuremath{\equiv}\text{Cov}[X(1),X(t+1)]/\text{Var}[X(1)]$. Furthermore, $z(t)={\ensuremath{\sum}}_{s=1}^{t}X(s)/t$ represents the successive proportions of red balls in an urn to which, at the $(t+1)\mathrm{th}$ stage, a red ball is added $[X(t+1)=1]$ with probability $q[z(t)]=(tanh{J[2z(t)\ensuremath{-}1]+h}+1)/2,\phantom{\rule{0.28em}{0ex}}J\ensuremath{\ge}0$, and a blue ball is added $[X(t+1)=0]$ with probability $1\ensuremath{-}q[z(t)]$. 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 $q(z)$. $C(t)\ensuremath{\sim}c+{c}^{\ensuremath{'}}\ifmmode\cdot\else\textperiodcentered\fi{}{t}^{l\ensuremath{-}1}$ with $c=0\phantom{\rule{0.28em}{0ex}}(>0)$ for $J<{J}_{c}\phantom{\rule{0.28em}{0ex}}(J>{J}_{c})$, and $l$ is the (larger) value of the slope(s) of $q(z)$ at the stable fixed point(s). On the boundary $J={J}_{c}(h),\phantom{\rule{0.16em}{0ex}}C(t)\ensuremath{\simeq}c+{c}^{\ensuremath{'}}\ifmmode\cdot\else\textperiodcentered\fi{}{(lnt)}^{\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}}$ and $c=0\phantom{\rule{0.28em}{0ex}}(c>0),\phantom{\rule{0.28em}{0ex}}{\ensuremath{\alpha}}^{\ensuremath{'}}=1/2\phantom{\rule{0.28em}{0ex}}(1)$ for $h=0\phantom{\rule{0.28em}{0ex}}(h\ensuremath{\ne}0)$. The system shows a continuous phase transition for $h=0$ and $C(t)$ behaves as $C(t)\ensuremath{\simeq}{(lnt)}^{\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}}g[(1\ensuremath{-}l)lnt]$ with a universal function $g(x)$ and a length scale $1/(1\ensuremath{-}l)$ with respect to $lnt$. $\ensuremath{\beta}={\ensuremath{\nu}}_{||}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\alpha}}^{\ensuremath{'}}$ holds with $\ensuremath{\beta}=1/2$ and ${\ensuremath{\nu}}_{||}=1$.
Source: Physical Review E
Authors: Shintaro Mori, Masato Hisakado
Institution: Kitasato University
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1783419441