Improved bounds for connection probabilities in random loop models

Abstract

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