noncolliding

Noncolliding refers to a property of a system of moving particles in which no two particles occupy the same position at any time. The term applies to both continuous-space models, such as Brownian motion or diffusion, and discrete models, such as random walks on graphs or lattices. In many constructions, noncolliding behavior is achieved by conditioning independent processes on the event of never colliding or by introducing a repulsive interaction that becomes infinite at zero distance.

A canonical continuous-time realization is noncolliding Brownian motion, also known in certain contexts as Dyson Brownian motion. Here, multiple particle trajectories evolve with a repulsive interaction that prevents collisions. A fundamental way to construct these processes is through Doob’s h-transform using the Vandermonde determinant, which conditions the system to maintain a strict order and hence avoid collisions. In the spectral interpretation, the eigenvalues of Hermitian matrix-valued Brownian motion form a family of noncolliding particles with dynamics that include a repulsive drift term. The joint distribution of the positions at fixed times often forms a determinantal point process and connects to random matrix theory, particularly the Gaussian unitary ensemble.

Discrete counterparts include noncolliding random walks and nonintersecting lattice paths. The Lindström–Gessel–Viennot lemma expresses the count and correlation structure of such path families in terms of determinants, linking noncolliding models to combinatorics and symmetric functions.

Noncolliding models are central in probability and mathematical physics for studying repulsion, eigenvalue statistics, and universality phenomena, and they provide a bridge between stochastic processes, random matrix theory, and combinatorics.