First Passage in Conical Geometry and Ordering of Brownian Particles

We survey recent results on first-passage processes in unbounded cones and their applications to ordering of particles undergoing Brownian motion in one dimension. We first discuss the survival probability S(t) that a diffusing particle, in arbitrary spatial dimension, remains inside a conical domain up to time t. In general, this quantity decays algebraically S ∼ t in the long-time limit. The exponent β depends on the opening angle of the cone and the spatial dimension, and it is root of a transcendental equation involving the associated Legendre functions. The exponent becomes a function of a single scaling variable in the limit of large spatial dimension. We then describe two first-passage problems involving the order of N independent Brownian particles in one dimension where survival probabilities decay algebraically as well. To analyze these problems, we identify the trajectories of the N particles with the trajectory of one particle in N dimensions, confined to within a certain boundary, and we use a circular cone with matching solid angle as a replacement for the confining boundary. For N = 3, the confining boundary is a wedge and the approach is exact. In general, this “cone approximation” gives strict lower bounds as well as useful estimates for the first-passage exponents. Interestingly, the cone approximation becomes asymptotically exact when N → ∞ as it predicts the exact scaling function that governs the spectrum of first-passage exponents.