Uniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular tree

<jats:p>We show that the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline1. png" /> <jats:tex-math> $4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-state anti-ferromagnetic Potts model with interaction parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline2. png" /> <jats:tex-math> $w\in (0,1)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on the infinite <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline3. png" /> <jats:tex-math> $(d+1)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular tree has a unique Gibbs measure if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline4. png" /> <jats:tex-math> $w\geq 1-\dfrac{4}{d+1_{_{\;}}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline5. png" /> <jats:tex-math> $d\geq 4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This is tight since it is known that there are multiple Gibbs measures when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline6. png" /> <jats:tex-math> $0\leq w\lt 1-\dfrac{4}{d+1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline7. png" /> <jats:tex-math> $d\geq 4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We moreover give a new proof of the uniqueness of the Gibbs measure for the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline8. png" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-state Potts model on the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline9. png" /> <jats:tex-math> $(d+1)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular tree for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline10. png" /> <jats:tex-math> $w\geq 1-\dfrac{3}{d+1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline11. png" /> <jats:tex-math> $d\geq 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline12. png" /> <jats:tex-math> $w\in (0,1)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000207_inline13. png" /> <jats:tex-math> $d=2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. </jats:p>