Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing.