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<p>We consider infinite random planar maps decorated by the critical Fortuin-Kasteleyn model with parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(q&gt;4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>&gt;</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. The paper demonstrates that when appropriately rescaled, these maps converge in law to the infinite continuum random tree as pointed metric-measure spaces, that is, with respect to the local Gromov-Hausdorff-Prokhorov topology. Furthermore, we also show that these maps do not admit any Fortuin-Kasteleyn loops with a macroscopic graph distance diameter. Our proof is based on Scott Sheffield’s hamburger-cheeseburger bijection.</p>

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Triviality of critical Fortuin-Kasteleyn decorated planar maps for \(q>4\)

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  • 发表日期：

    2025年11月1日

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  • Yuyang Feng

摘要

We consider infinite random planar maps decorated by the critical Fortuin-Kasteleyn model with parameter \(q>4\) q > 4 . The paper demonstrates that when appropriately rescaled, these maps converge in law to the infinite continuum random tree as pointed metric-measure spaces, that is, with respect to the local Gromov-Hausdorff-Prokhorov topology. Furthermore, we also show that these maps do not admit any Fortuin-Kasteleyn loops with a macroscopic graph distance diameter. Our proof is based on Scott Sheffield’s hamburger-cheeseburger bijection.