Persistency studies of trails and silhouettes on square and triangular lattices

The authors study the persistencies of trails and silhouettes on two-dimensional square and triangular lattices. These persistencies are induces by fixing the first step of the walk in the positive x direction. Trails are walks which are not allowed to overlap on themselves but may self-intersect whereas silhouettes are just the shadows of trails. Associated with each self-intersection is an attractive interaction energy epsilon =- mod epsilon mod . Thus a fugacity factor (f(I, theta ) theta =e It kBT/=e I theta , where I is the number of intersections) may be defined. This has the advantage of providing a handle on the temperature and such dependence of persistencies on temperature is studied for the first time in this paper. They are found to obey a law that may be expressed as a function of the critical exponents: (X 2k+1 l ( theta )) approximately l pk nu ( theta ) f(l) where l is the chain length, nu is the correlation exponent, f(l) approximately log e l or some weak function of l (e.g. l w where w<<1.0), p is a parameter and k=0, 1, 2.