Harmonic Measure and Winding of Conformally Invariant Curves

Abstract

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambdalnr. It obeys the scaling law f(alpha,lambda)=(1+lambda(2))f(alpha)-blambda(2) with alpha=alpha/(1+lambda(2)) and b=(25-c)/12, and where f(alpha) identical with f(alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to stochastic Löwner evolution.

2D growth processes: SLE and Loewner chains

“…The multifractal spectrum of harmonic measures of boundaries of critical clusters has been predicted by B. Duplantier in a serie of papers [54,55,56,57] using arguments based on applications of 2D gravity and of the KPZ formula [81] for gravitationally dressed dimensions. See the reviews [58,59].…”

Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

“…• (81) is the Coulomb gas formula obtained a long time ago for the exponent governing the winding angle distribution of a star made of a number L {h} of random paths (SLEs) [29,26,28]; • L {h} (82) represents, in a star topology, the effective number of SLEs that are exactly equivalent through QG to a collection of h i , i = 1, · · · , n, Brownian paths (which represent the set of powers h i of the harmonic measure), in addition to the n original SLEs [27,28];…”

“…Those concerned the multifractal spectra associated with the moments of the harmonic measure, i.e., the electrostatic potential, near CI curves, which could be derived exactly within that approach for any value of the central charge c [24]. A generalization concerned the peculiar mixed multifractal spectrum that describes both the harmonic measure moments and the indefinite winding or rotation (i.e., as logarithmic spirals) [25] of a CI curve about any of its points or of the Green lines of the potential [26]. Higher multifractality spectra were also introduced and calculated, concerning multiple moments of the harmonic measure and winding in various sectors of multiple random (SLE) paths in a star configuration [27,28].…”

“…The critical exponentx(h, p) depends on the two parameters h and p explicitly [26]. It is also interesting to consider in general multiple SLE paths with a star topology, and to define in a similar way the multiple mixed moments [28]:…”

Boundary behavior of SLE

“…Using arguments of conformal field theory, B. Duplantier and I. Binder derived the mixed multifractal spectrum f mixed (α, λ) for the scaling and winding (with respect to harmonic measure) in terms of the central charge: 2 , where b = (25 − c)/12. See [2] for definitions and [9] for more details of the results.…”