Yang edge singularity, and to the Baxter's models of hard-square lattice gas with next- .... ofz =z:(L,M) the free energy per site of the model has a power law sin-.
THE DIRECTED ANIMAES AND RELATED PROBLEm,S
Deepak DHAR Laboratoire
de Physique Th~orique e t Hautes E n e r g i e s
Universit6
P i e r r e e t Marie C u r i e
4, Place J u s s i e u - 75230 PARIS Cedex 05 and Tata I n s t i t u t e
o f Fundamental
Research
Homi Bhabha Road, Bombay, 400005,
INDIA
The directed animals problem is one of the simplest of the many problems in s t a t i s t i c a l mechanics, characterized by an absence of r e f l e c t i o n symmetry. Although this asymmetry, in the case of time-inversion, has been the central problem of nonequilibrium mechanics since the time of Boltzmann, the study of l a t t i c e models incorporating a preferred direction, and u t i l i s a t i o n of the powerful techniques of e q u i l i brium s t a t i s t i c a l mechanics to gain insight into non-equilibrium problems is much more recent. In t h i s a r t i c l e , I shall describe the directed animals problem as a prototype of such models, and describe i t s relationship to crystal growth models, the LeeYang edge s i n g u l a r i t y , and to the Baxter's models of hard-square l a t t i c e gas with nextnearest neighbour interactions.
1) Directed Animals Consider the two dimensional square lattice for simplicity. A directed animal of size n is defined as a connected set of n "occupied" sites, including the origin, such that for each occupied site (x,y) other than the origin, at least one ef the sites ( x - l , y) or (x, y - l ) is also occupied. Generalization of this definition to other lattices and dimensions is straightforward.
For large n, the total number distinct
animals with n sites An, their average transverse and longitudinal diameters
