Random Walk First Passage & Hitting Times

First passage time distribution for 1D random walks — Lévy-stable tail, mean FPT diverges for unbiased walk

Bias p (p>0.5 = drift right) 0.50

Target distance L 20

Walkers per batch 100

Max steps cutoff 3000

Sample count0

Mean FPT—

Median FPT—

Theory mean—

Fraction hit—

First passage time T is the first time a random walk starting at x=0 reaches x=L. For unbiased walk (p=0.5): P(T=t) ~ t^(-3/2) (Lévy-Smirnov distribution), mean FPT diverges. With bias p≠0.5: mean FPT = L/(2p-1) if p>0.5, infinite if p≤0.5. The heavy tail means rare long journeys dominate statistics.