An active particle in a confined domain — say, a bacterium in a microfluidic channel with two exits — must eventually escape through one exit or the other. The splitting probability is the probability of escaping through a specific exit. For passive Brownian particles, splitting probabilities are determined by the harmonic measure — the solution to Laplace's equation with boundary conditions set by the exits. The particle has no preference; geometry alone determines the probability.
Active particles have self-propulsion. They swim in a persistent direction, rotating slowly due to noise. The persistence length — how far the particle travels before randomizing its direction — introduces a new scale that competes with the domain geometry. When the persistence length is much smaller than the domain, the particle behaves like a Brownian walker and splitting probabilities match the harmonic measure. When the persistence length is comparable to or larger than the domain, the particle's initial orientation matters and splitting probabilities deviate from the passive prediction.
The paper derives exact expressions for splitting probabilities of confined active particles in specific geometries, revealing how the competition between persistence and confinement reshapes the escape statistics. The most striking result: for highly persistent particles, the splitting probability becomes nearly deterministic — the particle escapes through whichever exit its initial orientation points toward, regardless of which exit is geometrically closer.
Activity converts a geometric problem into a kinematic one. The passive particle asks “which exit is closer?” The active particle asks “which exit am I aimed at?” Increasing activity doesn't randomize the outcome — it determinizes it, replacing geometric chance with directional certainty.