The Thermodynamic Limit of Extreme First-Passage Times

Abstract

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 $d_w$ 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.

Figures (14)

• Figure 1: Convergence of the exact PDF of the slowest first-passage time for a Brownian particle following the diffusion Eq. (\ref{['eq:FP']}) to the Gumbel probability density function, given by Eq. (\ref{['eq:gumbelpdf']}), as $N$ increases. The results are shown for a fixed system size $L = 1000$ as a function of $z = 4\,\mathcal{T}/\pi^2L^2 - \log 2x_0\rho$, see also Appendix \ref{['append1a']}. The dashed line represents the Gumbel density function, while the symbols denote the exact results and the solid lines the theory developed in this paper, which will be discussed in detail later, see Eqs. (\ref{['eq:newzdef']},\ref{['eq:theorys']}). Here we used $D=1/2$ and $x_0=1$. As we increase $N$, the convergence to the asymptotic limit is clear, though deviations are visible for small $z$.
• Figure 2: Convergence of the exact PDF of the slowest first-passage time to the Fréchet density function, given by Eq. (\ref{['eq:frechetpdf']}), as the system size $L$ increases for fixed $N=10^3$. The dashed line represent the Fréchet probability density function, the symbols show the exact numerical solution, and the solid lines denotes the results from our theoretical framework, which will be discussed in detail later in Eqs. (\ref{['eq:newzdef']},\ref{['eq:theorys']}). The rescaled variable $z = D\,\pi\,\mathcal{T}/x_0^2 N^2$ (see also appendix \ref{['append1b']}), using parameters $D=1/2$ and $x_0=1$. Notably, the Fréchet distribution provides a good fit to the data when $L=10^5$, except at very large values of $z$, while for smaller system sizes, it does not accurately capture the behavior.
• Figure 3: PDF of the slowest FPT $\mathcal{T}$ for different regimes of $N$ and $L$ with $x_0=1$. The solid black line represents the theoretical prediction from Eqs. (\ref{['eq:surv1']},\ref{['eq:qapprox']}), and the symbols denote the exact PDF: (a) $N = L/x_0$: Comparison with both Gumbel and Fréchet shows that neither limit fully describes the behavior, with Gumbel fitting better at long times and Fréchet at short times. (b) High density limit $N \gg L/x_0$: The exact results are compared to Gumbel, which provides a good fit for large $N$ and finite $L$. (c) Low density limit $N \ll L/x_0$: The exact results are compared to Fréchet, which accurately captures the behavior in the large system limit with small $N$, provided that $\mathcal{T}$ is not too large. In all figures the thermodynamic limit we studied in Eqs. (\ref{['eq:surv1']},\ref{['eq:qapprox']}) matches nicely the exact results, with no fitting.
• Figure 4: Plot of $-\log{Q_N} / \rho$ as a function of the rescaled time $\mathcal{T} / L^2$ where the number of particles is $N = 10^4$. The figure demonstrates the scaling behavior of the cumulative probability for the slowest first-passage time in the thermodynamic limit, where $\rho = N / L$ remains fixed. The results validate the convergence to the theoretical form described by Eqs. (\ref{['eq:surv1']},\ref{['eq:qapprox']}). For small $\mathcal{T}/L^2$ we observe a power-law decay consistent with Fréchet statistics, while for large values, the distribution exhibits an exponential-like cutoff. The scaling function thus smoothly interpolates between these two regimes. Here we used $D=1/2$ and $x_0=1$.
• Figure 5: PDF of the slowest first-passage time: (Left) Results for different particle densities $\rho$ with $N = 2000$. The rescaled variable $z$ is defined via Eq. (\ref{['eq:newzdef']}). The solid line represents the theoretical prediction from Eqs. (\ref{['eq:newzdef']}-\ref{['eq:theorypdf']}), while the symbols denote the exact rescaled solution obtained from Eqs. (\ref{['eq:defq']},\ref{['eq:exactp']}). (Right) Results for different values of $N$ at fixed density $\rho = 1$, as we increase $N$ keeping $\rho$ fixed we reach the limiting law in Eq. (\ref{['eq:theorypdf']}). The solid black line represents the theoretical prediction from Eqs. (\ref{['eq:theorys']}) and (\ref{['eq:theorypdf']}), and the symbols show the exact rescaled solution. Here we used $x_0 = 1$ and $D = 1/2$. The results illustrate the emergence of a new limiting distribution whose shape is distinct from the Gumbel and Fréchet forms shown in Fig. \ref{['fig:comparevst']}, underscoring the different structure of the finite-density regime.