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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.21409 |
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| author | Janssen, A. J. E. M. |
| author_facet | Janssen, A. J. E. M. |
| contents | We consider in this work the crucial quantity $t_c$ that determines the critical inverse temperature $β_c$ in the $q$-state Potts model on sparse rank-1 random graphs where the vertices are equipped with a Pareto weight density $(τ-1)\,w^{-τ}\,{\cal X}_{[1,\infty)}(w)$. It is shown in \cite{ref1} that this $t_c$ is the unique positive zero of a function ${\cal K}$ 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 $q\geq3$ and $τ\geq4$, 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 ${\cal K}'$ and ${\cal K}''$ also have a unique positive zero, $t_c'$ and $t_c''$, respectively, and $t_c'=t_b$ and $t_c''=t_{\ast}$, where $t_b$ and $t_{\ast}$ are the unique positive zeros of ${\cal F}_0(t)-t\,{\cal F}_0'(t)$ and ${\cal F}_0''(t)$, respectively. Here, ${\cal F}_0(t)=E\,[W(e^{tW}-1)/(E\,[W]\,(e^{tW}+q-1))]$, 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, $γ_c=\exp(β_c)-1$ and $t_c$ are related according to $γ_c=t_c/{\cal F}_0(t_c)$.
We analyse $t_c$, $t_c'$ and $t_c''$ for general real $τ\geq4$ and general real $q>2$ by an appropriate formulation of their defining equations ${\cal K}(t_c)={\cal K}'(t_c')={\cal K}''(t_c'')=0$. Thus we find, along with the inequality $0<t_c''<t_c'<t_c<\infty$, the simple upper bounds $t_c<2\,{\rm ln}(q-1)$, $t_c'<\frac32\,{\rm ln}(q-1)$, $t_c''<{\rm ln}(q-1)$, as well as certain sharpenings of these simple bounds and counterparts about the large-$q$ behaviour of $t_c$, $t_c$ and $t_c''$. We show that these bounds are sharp in the sense that they hold with equality for the limiting homogeneous case $τ\to\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_21409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights Janssen, A. J. E. M. Mathematical Physics Probability We consider in this work the crucial quantity $t_c$ that determines the critical inverse temperature $β_c$ in the $q$-state Potts model on sparse rank-1 random graphs where the vertices are equipped with a Pareto weight density $(τ-1)\,w^{-τ}\,{\cal X}_{[1,\infty)}(w)$. It is shown in \cite{ref1} that this $t_c$ is the unique positive zero of a function ${\cal K}$ 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 $q\geq3$ and $τ\geq4$, 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 ${\cal K}'$ and ${\cal K}''$ also have a unique positive zero, $t_c'$ and $t_c''$, respectively, and $t_c'=t_b$ and $t_c''=t_{\ast}$, where $t_b$ and $t_{\ast}$ are the unique positive zeros of ${\cal F}_0(t)-t\,{\cal F}_0'(t)$ and ${\cal F}_0''(t)$, respectively. Here, ${\cal F}_0(t)=E\,[W(e^{tW}-1)/(E\,[W]\,(e^{tW}+q-1))]$, 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, $γ_c=\exp(β_c)-1$ and $t_c$ are related according to $γ_c=t_c/{\cal F}_0(t_c)$. We analyse $t_c$, $t_c'$ and $t_c''$ for general real $τ\geq4$ and general real $q>2$ by an appropriate formulation of their defining equations ${\cal K}(t_c)={\cal K}'(t_c')={\cal K}''(t_c'')=0$. Thus we find, along with the inequality $0<t_c''<t_c'<t_c<\infty$, the simple upper bounds $t_c<2\,{\rm ln}(q-1)$, $t_c'<\frac32\,{\rm ln}(q-1)$, $t_c''<{\rm ln}(q-1)$, as well as certain sharpenings of these simple bounds and counterparts about the large-$q$ behaviour of $t_c$, $t_c$ and $t_c''$. We show that these bounds are sharp in the sense that they hold with equality for the limiting homogeneous case $τ\to\infty$. |
| title | Analysis of quantities determining the critical inverse temperature in the annealed Potts model with Pareto vertex weights |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2508.21409 |