Z. Phys. B - Condensed Matter 57, 5%63 (1984) Condensed Zeitschrift Matter f~r Physik B 9 Springer-Verlag 1984 Radii of Gyration of Fully and Partially Directed Lattice Animals V. Privman* and M. Barma** Baker Laboratory, Cornell University, Ithaca, New York, USA Received June 7, 1984 Longitudinal and transverse radii of gyration are introduced for directed problems. Series analyses of these quantities to orders N--19 and 16, for fully and partially directed site lattice animals, respectively, on the square lattice yield the estimate vii =0.8177+0.0012, and suggest nonanalytic corrections in the partially directed case, with exponent 0 ~ 0.68. 1. Introduction : Axes and Radii of Directed Clusters The radius of gyration provides a natural measure of cluster size for undirected cluster statistics problems [1]. However, the generalization of this measure for directed problems has not been discussed: rather, spanning or 'caliper' diameters have been used [21. There are two difficulties that arise in using span- ning lengths: (a) in order to measure appropriate longitudinal and transverse dimensions one must know accurately the direction of the 'directed' axis of cluster growth. In the absence of a spatial sym- metry this axis may not coincide with a simple lat- tice direction: e.g., for directed percolation in d=2 with unequal bond probabilities Px and py. (b) Span- ning diameters have additional corrections to scaling [3] which are normally not expected for radii of gyration and their presence makes the estimation of v II and v• by series analysis techniques more difficult [2], [3]. In the rest of this sect., we will discuss appropriate definitions of axes and radii of directed clusters. In Sect. 2, we report series analyses for directed lattice animals in d--2. Let us first address the problem of locating the directed axis for anisotropic N-site clusters. For a given N-site cluster, principal axes can be defined, as usual in mechanics [4] by diagonalizing the inertia * From Sept. 84 at Dept. of Physics, California Institute of Technology, Pasadena, CA 91125, USA ** On leave from Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India tensor I~3(i,j=x, y in d=2). In d=2, the coordinate system defined by the principal axes [-4] is rotated by an angle 0 with respect to the X Y system (gener- alization to d>2 is straightforward). An average angle (ON), over all N-site clusters, defines the ap- proximate direction of the anisotropic axis. The se- quence <0N) obtained by studying clusters of N =2, 3.... sites can then be extrapolated by conven- tional methods to obtain an estimate of <0oo>. We 2 then define radii of gyration <]/~N, II) and <I//~N, =) by R2 1 N where ~ runs over the N sites, with coordinates X,,~ and X• (c~= 1, ...,N) in a system rotated by <00o), and 1 s AXII ~=XH,,---- ~, X[I, B, ' N~=I 1 N (2) --- ~, X• AX• N ~=I Moments of inertia of a single N-site cluster, when properly normalized and averaged (over all N-site clusters) can also be used as measures of the mean- squared longitudial and transverse cluster sizes.