Fix a dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be a random d-dimensional determinantal hypertree on n vertices. We prove that <Equation ID="Equ13"> <EquationSource Format="TEX">$\begin{aligned} \frac{\log |H_{d-1}(T_n,\mathbb {Z})|}{{{n\atopwithdelims (){d}}}} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfrac> <mrow> <mrow> <mo>log</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mi>n</mi> <mi>d</mi> </mfrac> </mfenced> </mfrac> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>converges in probability to a constant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation>, which satisfies <Equation ID="Equ14"> <EquationSource Format="TEX">$\begin{aligned} \frac{1}{2} \log \left( \frac{d+1}{e}\right) \le c_d\le \frac{1}{2} \log \left( d+1\right) . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>log</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>e</mi> </mfrac> </mfenced> <mo>≤</mo> <msub> <mi>c</mi> <mi>d</mi> </msub> <mo>≤</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>log</mo> <mfenced close=")" open="("> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>

The homology torsion growth of determinantal hypertrees

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2026年5月7日

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András Mészáros

摘要

Fix a dimension $d\ge 2$ $d \geq 2$ , and let $T_n$ $T_{n}$ be a random d-dimensional determinantal hypertree on n vertices. We prove that $\begin{aligned} \frac{\log |H_{d-1}(T_n,\mathbb {Z})|}{{{n\atopwithdelims (){d}}}} \end{aligned}$ $\begin{matrix} \frac{log | H_{d - 1} (T_{n}, Z) |}{(\binom{n}{d})} \end{matrix}$ converges in probability to a constant $c_d$ $c_{d}$ , which satisfies $\begin{aligned} \frac{1}{2} \log \left( \frac{d+1}{e}\right) \le c_d\le \frac{1}{2} \log \left( d+1\right) . \end{aligned}$ $\begin{matrix} \frac{1}{2} log (\frac{d + 1}{e}) \leq c_{d} \leq \frac{1}{2} log (d + 1) . \end{matrix}$