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arXiv · 1401.4793

Analysis of mean cluster size in directed compact percolation near a damp wall

H Lonsdale·I Jensen·J W Essam·A L Owczarek
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Abstract

We investigate the behaviour of the mean size of directed compact percolation clusters near a damp wall in the low-density region, where sites in the bulk are wet (occupied) with probability $p$ while sites on the wall are wet with probability $p_w$. Methods used to find the exact solution for the dry case ($p_w=0$) and the wet case ($p_w=1$) turn out to be inadequate for the damp case. Instead we use a series expansion for the $p_w=2p$ case to obtain a second order inhomogeneous differential equation satisfied by the mean size, which exhibits a critical exponent $γ=2$, in common with the wet wall result. For the more general case of $p_w=rp$, with $r$ rational, we use a modular arithmetic method of finding ODEs and obtain a fourth order homogeneous ODE satisfied by the series. The ODE is expressed exactly in terms of $r$. We find that in the damp region $0<r<2$ the critical exponent $γ^{\rm damp}=1$, in common with the dry wall result.

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H Lonsdale, I Jensen, J W Essam, A L Owczarek. 2014-01-20. Analysis of mean cluster size in directed compact percolation near a damp wall. https://doi.org/10.1088/1742-5468%2F2014%2F03%2Fp03004

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