1999/08/22 by Bertrand Duplantier · 164 citations
Materials Science · Mathematics · Physics and Astronomy · #Covariant transformation #Fractal #Hausdorff dimension #Invariant (physics) #Material Dynamics and Properties #Mathematical analysis #Mathematical physics #Mathematics #Multifractal system #Phase transition #Physics #Potts model #Quantum mechanics #Statistical Mechanics and Entropy #Theoretical and Computational Physics #Wedge (geometry) #cond-mat.stat-mech #math-ph #math.MP #math.PR
paper · pdf · doi:10.1103/physrevlett.84.1363
published in Physical Review Letters 84(7), 1363-1367 (American Physical Society) · 5 pages, 1 figure
arxiv created 1999/08/22 · openalex publication_date 2000/02/14 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a Q-state Potts cluster, is solved in two dimensions. The dimension \stackrel^f(\ensuremathθ) of the boundary set with local wedge angle \ensuremathθ is \stackrel^f(\ensuremathθ)\phantom\rule0ex0ex=\phantom\rule0ex0ex\frac\ensuremathπ\ensuremathθ\ensuremath-\frac25\ensuremath-c12\frac(\ensuremathπ\ensuremath-\ensuremathθ)2\ensuremathθ(2\ensuremathπ\ensuremath-\ensuremathθ), with c the central charge of the model. As a corollary, the dimensions DEP of the external perimeter and DH of the hull of a Potts cluster obey the duality equation (DEP\ensuremath-1)(DH\ensuremath-1)\phantom\rule0ex0ex=\phantom\rule0ex0ex(1)/(4). A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.