This paper investigates translation-invariant p-adic generalized Gibbs measures for the p-adic Ising model with a homogeneous external field on a Cayley tree of order three, assuming <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9516_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p > 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We demonstrate that if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9516_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$p \equiv 1 (\operatorname {mod} {6})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mo>mod</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, then there exist four translation-invariant p-adic generalized Gibbs measures; if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9516_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$p \not \equiv 1 (\operatorname {mod} {6})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≢</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mo>mod</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, there exist exactly two. Additionally, for any prime <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9516_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p > 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the occurrence of a phase transition in this model.

Translation-invariant p-adic generalized Gibbs measures for the Ising model with a homogeneous external field on a Cayley tree

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发表日期：

2025年7月30日

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Zulxumor Abdukaxorova

摘要

This paper investigates translation-invariant p-adic generalized Gibbs measures for the p-adic Ising model with a homogeneous external field on a Cayley tree of order three, assuming $p > 3$ $p > 3$ . We demonstrate that if $p \equiv 1 (\operatorname {mod} {6})$ $p \equiv 1 (mod 6)$ , then there exist four translation-invariant p-adic generalized Gibbs measures; if $p \not \equiv 1 (\operatorname {mod} {6})$ $p ≢ 1 (mod 6)$ , there exist exactly two. Additionally, for any prime $p > 3$ $p > 3$ , we establish the occurrence of a phase transition in this model.