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.
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.