Size and shape of directed lattice animals
J. Phys. A: Math. Gen. 15 (1982) L177-L187. Printed in Great Britain，-0001，（）：
We use the position-space renormalisation group and exact enumeration methods to study configurational properties of directed lattice animals. These are clusters made up of directed bonds in which the'tail'of a new bond must be added to the 'head' of already existing bonds. Furthermore, the directed bonds have a net orientational order with respect to some preferred anisotropy axis. In the limit that the number of bonds N+CQ, directed animals become extremely anisotropic and two independent correlation lengths, one parallel and one perpendicular to the preferred axis, are required to describe their shape. Mean-field theory suggests that these lengths diverge with exponents of Y I I=f and vl=$ respectively. Below seven dimensions mean-field theory breaks down, and we analyse our enumeration data to estimate the location of the critical point, the exponents YII and v l, and the exponent 0 characterising the singularity of the directed animal generating function. In two dimensions, we estimate that YH=0.8, vl=0.5 and 0=0.5, and our estimates interpolate smoothly with dimension to their respective mean-field limits. In two dimensions, we also apply a one-parameter position-space renormalisation group using small cells which gives reasonable estimates for the location of the critical point and q. In addition, we formulate a two-parameter renormalisation which allows us to study animals with a variable fraction of directed bonds, and thereby describe the crossover between directed and isotropic animals. We find that any anisotropy ensures that the critical behaviour belongs to the universality class of directed animals.
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