The Poisson approximation theorem states that when $m$ balls are tossed uniformly and randomly into $n$ bins, the distribution of the bin loads can sometimes be approximated by $n$ independent Poisson random variables of mean $m/n$. We show that when the balls are tossed via a random $\mathbb{F}_2$-linear map, the load distribution is still governed by this Poisson distribution, albeit in a (necessarily) weaker manner. This allows us to show that a random linear map behaves like a fully random map in the following ways:

1. Expected loads. We show that the expected value of the $r$th largest bin is governed by the $r$th largest value among $n$ independent Poisson random variables of mean $m/n$. Specializing to the canonical setting of $m=n$ and $r=1$, our bounds show that the expected maximum load when $n$ balls are hashed into $n$ bins by a random $\mathbb{F}_2$-linear map is no more than the corresponding expectation for a fully random map plus $\frac{3}2 + o(1)$, drastically improving on bounds derived by the framework of Jaber, Zuckerman, and the author (STOC 2025). Specializing to $m\gg n\log n$, we positively resolve a question of Pathegama and Barg (TIT 2025) over $\mathbb{F}_2$, who had asked whether the expected max load is $(1+o(1))m/n$ in this regime.

2. Concentration. When $m\gg n\log (1/\rho)$ balls are tossed into $n$ bins randomly, we expect with $0.99$ probability that every density-$\rho$ subset of bins will contain a $(1\pm o(1) )\rho$-fraction of the balls. We show that this remains the case if the $m$ balls are tossed using a random linear map, and our bound matches the $o(1)$-factor produced by the fully random case. This gives a stronger version of the Leftover Hash Lemma for linear maps. An implication of this is that all bins have load $m/n \pm O(\sqrt{(m/n)\log n})$ with probability $0.99$, which strictly improves the two-sided bounds of Dhar and Dvir (FOCS 2022) over $\mathbb{F}_2$, and quantitatively matches the order of deviation for constant failure probability of fully random hashing.

To obtain these results, we first show that the average of a log-potential on the bin loads is upper bounded by the cumulant generating function of a Poisson random variable of mean $m/n$. Expectation and tail bounds then easily fall out of this inequality, similar to how Chernoff tail bounds fall out of moment generating function inequalities.