A note on Ursell functions for the Blume-Capel model

RESUMO

We study the sign structure of Ursell functions in the Blume--Capel model. For $\Delta \le \log 2$, we give a short proof of the alternating sign property using an Ising representation. We then construct counterexamples for every $\Delta > \log 2$, showing that the condition $\Delta \le \log 2$ is sharp.