Fractal Structure of High-Temperature Graphs of O($N$) Models in Two Dimensions

The fractal structure and critical properties of the high-temperature graphs of the two-dimensional O($N)$ model close to criticality are investigated. Based on Monte Carlo simulations, De Gennes' results for polymer chains, corresponding to the limit $N \to 0$, are generalized to random loops for arbitrary $-2 \leq N \leq 2$. The loops are also studied close to their tricritical point, known as the $\Theta$ point in the context of polymers, where they collapse. The corresponding fractal dimensions are argued to be in one-to-one correspondence with those at the critical point, leading to an analytic prediction for the magnetic scaling dimension at the O($N)$ tricritical point.