The Tricritical Behaviour of Self–Interacting Partially Directed Walks

We present the thermodynamics of two variations of the Interacting Partially Directed Self Avoiding Walk problem by discussing versions where the length of the walks assume real as well as a integral values. While the discrete model has been considered previously to varying degrees of success the continuous model we now define has not. The exam-ination of the continuous model leads to the exact derivation of several exponents. For the discrete model some of these exponents can be calculated using a continued fraction representation. For both models the crossover exponent φ is found to be 2/3. Moreover we confirm the tricritical nature of the collapse transition in the generalised ensemble and calculate the full scaling form of the generating function. Additionally, the similarities noticed previously, but left unexplored, to other models are explained with the aid of necklacing arguments