Poisson limits for large boxes in sparse random allocations

Let $(Y_1,\dots,Y_n)$ be the random allocation of $m$ balls into $n$ boxes in which an allocation $(y_1,\dots,y_n)$ has probability proportional to $\prod_iw_{y_i}$, and let $\xi_1,\dots,\xi_n$ be independent with $\mathbb P(\xi_i=k)$ proportional to $w_k\tau^k$, where $\tau$ is chosen so that $\mathbb E\xi_i=m/n$. Janson showed that when $m/n$ converges to a positive limit below a threshold determined by the weights, the numbers of boxes with many balls are asymptotically Poisson and the largest boxes of the allocation are close in distribution to the largest of $\xi_1,\dots,\xi_n$, and he asked whether the same holds when $m/n\to0$. Let $a$ be the least $k\ge1$ with $w_k>0$. The answer is positive for every sequence with $m\to\infty$ if every $k$ with $w_k>0$ is a multiple of $a$, for instance if $w_1>0$, and it is negative along every sequence on which $m$ is bounded. If some $k$ with $w_k>0$ is not a multiple of $a$, there are sequences with $m\to\infty$ and $m/n\to0$ along which the answer is negative. When the least such $k$, say $b$, is coprime to $a$, we characterize the sequences along which the answer is positive in terms of $n\mathbb P(\xi_1=b)$ and the residue of $m$ modulo $a$; when $n\mathbb P(\xi_1=b)$ has a positive finite limit, the number of boxes with at least $b$ balls converges to a Poisson distribution conditioned on a residue class. The proofs rest on an exact formula for the joint law of the boxes whose sizes are neither $0$ nor $a$ and on a local limit theorem for the binomial distribution with explicit constants.