O(n) loop-decorated random planar maps are a well-studied model coupling quantum gravity with statistical mechanics. An important progress in the study of their geometry was made by Borot, Bouttier and Guitter when they established that O(n) loop-decorated maps could be decomposed recursively by cutting the configurations along the loops. The central building block in this decomposition, called the gasket, is obtained by removing the outermost loops and their interiors. They discovered that for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, at criticality, the gaskets are random maps with high degrees, usually called stable maps. However the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> remained excluded. We prove that for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> the gaskets of critical rigid O(n) loop-decorated random planar maps are 3/2-stable maps. The case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> thus corresponds to the critical case in random planar maps. Contrary to the cases <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, a slowly varying function arises in the perimeter exponent. The proof relies on the Wiener–Hopf factorisation for random walks and is robust enough to deal with bipartite maps with arbitrarily large degrees. Our techniques also provide a characterisation of weight sequences of critical O(2) loop-decorated maps.

Gaskets of O(2) Loop-Decorated Random Planar Maps

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发表日期：

2025年12月2日

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Emmanuel Kammerer

摘要

O(n) loop-decorated random planar maps are a well-studied model coupling quantum gravity with statistical mechanics. An important progress in the study of their geometry was made by Borot, Bouttier and Guitter when they established that O(n) loop-decorated maps could be decomposed recursively by cutting the configurations along the loops. The central building block in this decomposition, called the gasket, is obtained by removing the outermost loops and their interiors. They discovered that for $n<2$ $n < 2$ , at criticality, the gaskets are random maps with high degrees, usually called stable maps. However the case $n=2$ $n = 2$ remained excluded. We prove that for $n=2$ $n = 2$ the gaskets of critical rigid O(n) loop-decorated random planar maps are 3/2-stable maps. The case $n=2$ $n = 2$ thus corresponds to the critical case in random planar maps. Contrary to the cases $n<2$ $n < 2$ , a slowly varying function arises in the perimeter exponent. The proof relies on the Wiener–Hopf factorisation for random walks and is robust enough to deal with bipartite maps with arbitrarily large degrees. Our techniques also provide a characterisation of weight sequences of critical O(2) loop-decorated maps.