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Hypergraph independence polynomials with a zero close to the origin
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abstract
For each uniformity $k \geq 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\lvert\lambda\rvert = O\left(\frac{\log \Delta}{\Delta}\right)$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.
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Lower tails for triangles inside the critical window
The lower-tail large deviation rate for triangle counts in the critical random graph is determined in closed form for part of the parameter plane, with phase transitions shown for small targets.
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