Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights

Abstract

We consider in this work the crucial quantity $t_c$ that determines the critical inverse temperature ${\beta }_c$ in the $q$-state Potts model on sparse rank-1 random graphs where the vertices are equipped with a Pareto weight density $(\tau -1)w^{-\tau }{\mathcal{X}}_{[1,\infty )}(w)$. It is shown in \cite{ref1} that this $t_c$ is the unique positive zero of a function $\mathcal{K}$ that is obtained by an appropriate combination of the stationarity condition and the criticality condition for the case the external field $B$ equals 0 and that $q\ge 3$ and $\tau \ge 4$, see \cite{ref1}, Theorem~1.14 and Theorem ~1.21 and their proofs in \cite{ref1}, Section~7.1 and Section~7.3. From the proof of \cite{ref1}, Theorem~1.14, it is seen that ${\mathcal{K}}^{\prime }$ and ${\mathcal{K}}^{\prime \prime }$ also have a unique positive zero, $t_c^{\prime }$ and $t_c^{\prime \prime }$, respectively, and $t_c^{\prime }=t_b$ and $t_c^{\prime \prime }=t_{\ast }$, where $t_b$ and $t_{\ast }$ are the unique positive zeros of ${\mathcal{F}}_0(t)-t{\mathcal{F}}_0^{\prime }(t)$ and ${\mathcal{F}}_0^{\prime \prime }(t)$, respectively. Here, ${\mathcal{F}}_0(t)=E[W(e^{tW}-1)/(E[W](e^{tW}+q-1))]$, and $t_c$, $t_b$ and $t_{\ast }$ play a key role in the graphical analysis of \cite{ref1}, Section~5.1 and Figure~1. Furthermore, ${\gamma }_c=\exp ({\beta }_c)-1$ and $t_c$ are related according to ${\gamma }_c=t_c/{\mathcal{F}}_0(t_c)$. We analyse $t_c$, $t_c^{\prime }$ and $t_c^{\prime \prime }$ for general real $\tau \ge 4$ and general real $q>2$ by an appropriate formulation of their defining equations $\mathcal{K}(t_c)={\mathcal{K}}^{\prime }(t_c^{\prime })={\mathcal{K}}^{\prime \prime }(t_c^{\prime \prime })=0$. Thus we find, along with the inequality $0<t_c^{\prime \prime }<t_c^{\prime }<t_c<\infty$, the simple upper bounds $t_c<2\ln (q-1)$, $t_c^{\prime }<\frac{3}{2}\ln (q-1)$, $t_c^{\prime \prime }<\ln (q-1)$, as well as certain sharpenings of these simple bounds and counterparts about the large-$q$ behaviour of $t_c$, $t_c$ and $t_c^{\prime \prime }$. We show that these bounds are sharp in the sense that they hold with equality for the limiting homogeneous case $\tau \to \infty$.