Year: 1999
Type: article
Abstract: $2\mathrm{D}$ percolation path exponents ${x}_{\ensuremath{\ell}}^{P}$ describe probabilities for traversals of annuli by $\ensuremath{\ell}$ nonoverlapping paths on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ models whose exponents, believed to be exact, yield ${x}_{\ensuremath{\ell}}^{P}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({\ensuremath{\ell}}^{2}\ensuremath{-}1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\ell}/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, ${D}_{\mathrm{EP}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{-}x}_{3}^{P}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4/3$, and also explains the absence of narrow gate fjords, which was originally noted by Grossman and Aharony.
Source: Physical Review Letters
Institutions: Princeton University, Tel Aviv University
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2031343032