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<p>The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. Typical pairs of points in the directed landscape are connected by a unique geodesic. However, there are exceptional pairs of points connected by more complicated geodesic networks. We show that, up to isomorphism, exactly 27 geodesic networks exist in the directed landscape. We also find Hausdorff dimensions in a scaling-adapted metric on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1355_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">↑</mo> <mn>4</mn> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{4}_{\uparrow }$</EquationSource> </InlineEquation> for the sets of endpoints of each of these networks.</p>

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The 27 geodesic networks in the directed landscape

  • 期刊论文

  • 发表日期：

    2025年8月14日

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  • Duncan Dauvergne

摘要

The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. Typical pairs of points in the directed landscape are connected by a unique geodesic. However, there are exceptional pairs of points connected by more complicated geodesic networks. We show that, up to isomorphism, exactly 27 geodesic networks exist in the directed landscape. We also find Hausdorff dimensions in a scaling-adapted metric on R ↑ 4 $\mathbb{R}^{4}_{\uparrow }$ for the sets of endpoints of each of these networks.