Maximal flow through thin boxes with heavy-tailed capacities

Kesten's law of large numbers for the maximal flow $\Phi$ from the bottom to the top of a box $[0,k]\times[0,l]\times[0,m]$ in $\mathbb{Z}^3$ with i.i.d. capacities assumes a finite exponential moment, and Auffinger, Damron and Hanson ask whether the limit of $\Phi/(kl)$ exists without it. We show that without a replacement moment condition it need not. For every height function $m(N)$ with $\liminf m(N)/\log N=0$ there is a capacity law with no atom at $0$ such that $\limsup\Phi/N^{d-1}=+\infty$ almost surely in the boxes $[0,N]^{d-1}\times[0,m(N)]$ of $\mathbb{Z}^d$, $d\ge2$. The heights may tend to infinity, for instance as $(\log N)^{1/2}$, and so lie inside Kesten's range. For heights with $m(N)/\log N\to\infty$, Rossignol and Théret proved convergence to a finite flow constant for every capacity law; for the laws constructed here the limit therefore depends on how fast the height grows.