Improved bounds for connection probabilities in random loop models

We revisit and extend results by Ueltschi [19] on the application of reflection positivity to loop models with $\theta \in {\mathbb{N}}_{\ge 2}$. By exploiting additional flexibility in the method, we prove the existence of long loops over a broader range of parameters $u$ and $\theta$, and establish new lower bounds for connection probabilities and the critical parameter ${\beta }_c$. Our results are compared with recent numerical simulations, providing further insight into the phase diagram of quantum spin systems.