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 Fréchet distribution in an infinite domain. Here, we study the physically relevant thermodynamic limit where both N and L diverge while the density $ρ= N/L$ remains constant. We obtain an explicit solution for the extreme value in the thermodynamic limit, which recovers the Fréchet 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 dw and fractal dimension df 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.