<p>The (non-spanning) tree decorated quadrangulation is a random pair consisting of a quadrangulation and a subtree chosen uniformly over the set of pairs with a prescribed size. In this paper, we study the tree decorated quadrangulation in the critical regime: when the number <i>f</i> of faces of the map is proportional to the square of the size of the tree. We show that with high probability in this regime, the diameter of the tree lies between <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1369_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$o(f^{1/4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1369_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{1/4}/\log ^\alpha (f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">/</mo> <msup> <mo>log</mo> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1369_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Thus after scaling distances by <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1369_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{-1/4}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, the critical tree decorated quadrangulation converges to a Brownian disk whose boundary has been identified to a point. These results imply the triviality of the shocked map: the metric space generated by gluing a Brownian disk with a continuous random tree.</p>

On the triviality of the shocked map

期刊论文
发表日期：

2025年3月27日

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Luis Fredes,
Avelio Sepúlveda

摘要

The (non-spanning) tree decorated quadrangulation is a random pair consisting of a quadrangulation and a subtree chosen uniformly over the set of pairs with a prescribed size. In this paper, we study the tree decorated quadrangulation in the critical regime: when the number f of faces of the map is proportional to the square of the size of the tree. We show that with high probability in this regime, the diameter of the tree lies between $o(f^{1/4})$ $o (f^{1 / 4})$ and $f^{1/4}/\log ^\alpha (f)$ $f^{1 / 4} / {log}^{α} (f)$ , for all $\alpha >1$ $α > 1$ . Thus after scaling distances by $f^{-1/4}$ $f^{- 1 / 4}$ , the critical tree decorated quadrangulation converges to a Brownian disk whose boundary has been identified to a point. These results imply the triviality of the shocked map: the metric space generated by gluing a Brownian disk with a continuous random tree.