Extreme First-Passage Statistics in Stochastic Systems

• Extreme first-passage statistics are the study of the minimum or maximum arrival times in independent stochastic processes, offering a clear framework for quantifying rare events.
• This field integrates large-deviation theory, classical extreme-value analysis, and spectral methods to derive universal limiting laws and detailed scaling regimes.
• Its applications span diffusion, biochemical signaling, material failure, and network transport, providing actionable insights for modeling and experimental diagnostics.

Extreme first-passage statistics concerns the behavior of the extremal (minimum or maximum) values among a population of statistically identical, independent first-passage times to a specified event—such as the first occurrence of a rare event among many agents, the earliest arrival in transport processes, or the maximum displacement reached during escape. This area is central to quantifying high-end fluctuations and rare event timescales in diffusion, random walks, stochastic resetting, reaction kinetics, transport networks, and related many-body stochastic systems. Recent research synthesizes large-deviation analysis, classical extreme-value theory (EVT), probabilistic coupling, and spectral decompositions to systematically derive the statistics of extreme first-passage events, their universal limiting laws, scaling regimes, and dependence on geometry, injection protocol, underlying dynamics, and environmental heterogeneities.

1. Mathematical Framework and Paradigmatic Regimes

For $N$ independent identically distributed (i.i.d.) random variables $\tau_1,\dots,\tau_N$ representing first-passage times, the primary object is the extremal order statistics

$$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$

with cumulative distribution functions (CDFs)

$$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$

The corresponding probability densities are sharply peaked and governed by the extreme tails of the single-particle survival, $S(t) = P(\tau_1>t)$. The prevailing scaling regimes and the applicable extreme-value limiting laws depend sensitively on the small-$t$ (or large-$t$ for maxima) behavior of $S(t)$, the presence or absence of hard lower bounds, the injection profile, graph or spatial geometry, and the dynamical class.

The main asymptotic possibilities are:

• Gumbel regime: When $S(t)\sim 1-A t^p e^{-C/t}$ for $t\to 0^+$ (diffusive searchers, continuous state space), the centered and scaled minimum converges to the standard Gumbel law, and moments decay as $\tau_1,\dots,\tau_N$0 (Lawley, 2019, Lawley, 2019, Lawley et al., 2019).
• Weibull regime: If there is a hard minimal hitting time $\tau_1,\dots,\tau_N$1 with $\tau_1,\dots,\tau_N$2 and $\tau_1,\dots,\tau_N$3, and near $\tau_1,\dots,\tau_N$4 the probability of arrival follows $\tau_1,\dots,\tau_N$5 as $\tau_1,\dots,\tau_N$6, then $\tau_1,\dots,\tau_N$7 rescaled converges to a Weibull law with polynomial approach to $\tau_1,\dots,\tau_N$8—typical in piecewise-deterministic Markov systems, finite-speed searchers, and discrete networks (Lawley, 2019, Karamched, 7 Jan 2026).
• Fréchet regime: Algebraic survival probability tails $\tau_1,\dots,\tau_N$9 yield Fréchet laws, relevant for maximum statistics of very broad-tailed processes (Baravi et al., 7 Sep 2025).

For the fastest first-passage time in $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$0-dimensional Brownian search (no drift), the classical Lawley–Madrid–Schuss formula holds: $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$1 where $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$2 is the shortest-path distance and $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$3 the diffusivity (Lawley, 2019, Lawley et al., 2019). This scaling is universal for single-particle survival probabilities with an Arrhenius-type or rare-event exploding tail, independent of force field or geometry as long as the minimal distance is well-defined.

2. Rare Events, Large Deviations, and Extreme Pathways

For rare first-passage events—such as escape over potential barriers or weak-noise stochastic dynamics—the distribution of the single-walker first-passage time $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$4 satisfies a large-deviation principle: $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$5 with action functional $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$6 derived from a Freidlin–Wentzell rate function depending on system dynamics and boundary conditions (MacLaurin et al., 2023). For $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$7 such rare events in the limit $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$8, $$T_{1,N} = \min\{\tau_1,\dots,\tau_N\}, \qquad T_{N,N} = \max\{\tau_1,\dots,\tau_N\}$$9 with $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$0, the distribution of the fastest time $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$1 is

$$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$2

and the mean is set by the saddle-point: $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$3 The most-likely path corresponding to $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$4 is the solution to a Hamiltonian boundary-value problem: $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$5 enforcing a finite-time, dynamically-biased trajectory distinct from the single-walker TFS path (MacLaurin et al., 2023).

3. Generalizations: Spatial, Temporal, and Environmental Complexity

• Time-dependent and asynchronous injection: When particles are injected at random or with a spread over time, the effective survival function becomes a time-convolution: $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$6 and the scaling of the mean fastest time can transition from $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$7 (instantaneous injection) to $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$8 or even slower, determined by the sharper of the search or injection kernel small-time tails (Grebenkov et al., 24 Mar 2025).
• Random/inhomogeneous environments: In random walks in random environments (RWRE), extreme statistics encode both sampling fluctuations (Gumbel regime) and additional environmental fluctuations, whose variance contributes nontrivial power-law terms and is governed by a universality class determined by the variance of local drift fields (Hass et al., 2024, Hass et al., 2023). The variance scales as $$P(T_{1,N}>t) = [P(\tau_1>t)]^N, \qquad P(T_{N,N}\le t) = [P(\tau_1\le t)]^N.$$9 for first-passage times in 1d RWRE, revealing otherwise hidden environmental disorder.
• Networks, graphs, and ballistic universality classes: On discrete networks, the minimal arrival time to a target may be strictly bounded below by the geodesic (minimal steps). In "injection-limited" regimes, detours cost finite entropic penalties, the arrival-time law is "hard-edge" Poissonian, and classical extreme value distributions fail entirely (Karamched, 7 Jan 2026). On treelike or high-dimensional graphs with bulk-limited behavior, a breakdown of this penalty leads to classical Gumbel or Gaussian limiting laws.

4. Extreme-Value Statistics of Trajectory Functionals

A duality exists between first-passage statistics and temporal process extrema (maxima/minima of stochastic trajectories). For continuous ergodic Markov processes, the distribution function of the running maximum is equivalent to the first-passage survival to level $S(t) = P(\tau_1>t)$0 up to time $S(t) = P(\tau_1>t)$1, allowing spectral or renewal approaches to obtain the large-deviation tail and bounds on extremes (Hartich et al., 2019). For Brownian motion, resetting dynamics, and run-and-tumble processes, explicit formulas for the maximum $S(t) = P(\tau_1>t)$2 and time of maximum $S(t) = P(\tau_1>t)$3 pre-absorption have been obtained, often revealing nonmonotonicity and the existence of optimal resetting rates (Guo et al., 2023, Guo et al., 2023, Huang et al., 16 Jun 2025, Singh et al., 2022).

5. Statistical Inference, Fluctuations, and Diagnostics

In both small and large $S(t) = P(\tau_1>t)$4 (sample) regimes, the characterization of uncertainty in extremes requires concentration-of-measure techniques to bound deviations of empirical order statistics from population values. Nonasymptotic inequalities and two-sided "squeeze" bounds provide model-free error control over sample maxima and minima, crucial for robust inference in kinetic networks, experimental proteomics, and simulation (Bebon et al., 2023). Further, diagnostics such as the trajectory-to-trajectory uniformity index quantify sample-to-sample fluctuations; the transition from unimodal to bimodal distributions of uniformity values signals the breakdown of the mean as a faithful descriptor of extreme statistics (Mattos et al., 2013).

6. Applications and Physical Implications

Extreme first-passage statistics underlie timescales in:

• Biochemical and cell signaling, where the first molecule triggers a reaction and $S(t) = P(\tau_1>t)$5 can be orders of magnitude faster than $S(t) = P(\tau_1>t)$6 (Lawley et al., 2019, Lawley, 2019).
• Astrochemistry, materials fracture, and catalysis, where rare, fastest-activation determines macroscopic response.
• Stochastic search, parallel computation, and epidemiology, where redundancy (many independent searchers) amplifies the speed of discovery or transmission.
• Photon transport and optical media, where extremal arrival times (not diffusion-limited) probe index-averaged ballistic paths and reveal limits of conventional transport theory (Carroll-Godfrey et al., 26 Feb 2025).

Empirically, the statistics of extremal arrivals can be used as "microscopes" for otherwise unmeasurable environmental or topological disorder (Hass et al., 2024, Hass et al., 2023). In modeling and experiment, neglecting the proper extreme-statistics regime or injection protocol can lead to severe underestimation or mischaracterization of process timescales and fluctuations (Grebenkov et al., 24 Mar 2025).

7. Summary Table: Core Asymptotic Laws for Fastest First-Passage Times

Dynamics / Model	Extreme Law (leading scaling)	Universal Feature
Brownian diffusion	$S(t) = P(\tau_1>t)$7	Gumbel regime, logarithmic
Finite-speed PDMP	$S(t) = P(\tau_1>t)$8 (Weibull law; $S(t) = P(\tau_1>t)$9 Weibull)	Hard lower bound $t$0
Discrete networks	$t$1 hard-edge/Poisson statistics	Ballistic ballistic class
RWRE (environmental noise)	$t$2	Environmental variance dominates
Extended injection	$t$3	Injection tail controls scaling

The table summarizes universal scaling laws, emphasizing the dependence on tail class, geometry, and protocol (MacLaurin et al., 2023, Hass et al., 2024, Karamched, 7 Jan 2026, Lawley, 2019, Grebenkov et al., 24 Mar 2025).

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Extreme first-passage statistics thus unify the probabilistic, geometric, and large-deviation aspects of the fastest and rarest events in complex stochastic processes, yielding a framework that quantitatively controls acceleration, fluctuations, and path selection in both theoretical and experimental settings.