The Thermodynamic Limit of Extreme First-Passage Times

The statistics of the slowest first-passage time among a large population of $N$ searchers is crucial for determining the completion time of many stochastic processes. Classical extreme-value theory predicts that for diffusing particles in a finite domain of size $L$, the slowest first passage time follows a Gumbel distribution, but a Fr\'echet distribution in an infinite domain. Here, we study the physically relevant thermodynamic limit where both $N$ and $L$ diverge while the density $\rho = N/L$ remains constant. We obtain an explicit solution for the extreme value in the thermodynamic limit, which recovers the Fr\'echet and Gumbel distributions in the low- and high-density limits, respectively, and reveals new, nontrivial behavior at intermediate densities. We then extend the framework to compact diffusion on fractal domains, showing that the walk dimension $d_w$ and fractal dimension $d_f$ control the extreme-value statistics via geometry-dependent scaling. The theory yields the full set of moments and finite-density corrections, providing a unified description of slowest-arrival times in confined Euclidean and fractal media.