Bounds on the Mod 2 Homology of Random 2-Dimensional Determinantal Hypertrees

Abstract As a first step towards a conjecture of Kahle and Newman, we prove that if $$T_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> is a random 2-dimensional determinantal hypertree on n vertices, then $$\begin{aligned} \frac{\dim H_1(T_n,\mathbb {F}_2)}{n^2} \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfrac> <mml:mrow> <mml:mo>dim</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation principle for the Erdős–Rényi random graph by Chatterjee and Varadhan.