<p>In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3775_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{J}(k)=\beta \, k^{-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>J</mtext> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mspace width="0.166667em" /> <msup> <mi>k</mi> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3775_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, in the whole subcritical regime <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3775_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta <\beta _c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo><</mo> <msub> <mi>β</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for which the parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3775_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is greater than 3/2.</p>

Continuous eigenfunctions of the transfer operator for Dyson models

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2025年5月21日

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Anders Johansson,
Anders Öberg,
Mark Pollicott

摘要

In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions $\textrm{J}(k)=\beta \, k^{-\alpha }$ $J (k) = β k^{- α}$ , $k\ge 1$ $k \geq 1$ , in the whole subcritical regime $\beta <\beta _c$ $β < β_{c}$ for which the parameter $\alpha $ $α$ is greater than 3/2.