Interlacement

Interlacement denotes the state of being interlaced or intertwined. In mathematics, it is used to describe a crossing or weaving relation between families of curves, paths, or subspaces, and the term takes on specific technical meanings in different contexts.

One prominent mathematical use is random interlacements. On the lattice Zd (d ≥ 3), random interlacements is a probabilistic model introduced by Sznitman to describe the local limit of the trace left by a family of random walks on large finite graphs. It is constructed as a Poisson point process of doubly infinite trajectories modulo time shift, with intensity parameter u ≥ 0. The interlacement set I^u is the union of the traces of all trajectories; the vacant set V^u is its complement. For dimensions d ≥ 3, there is a nontrivial percolation transition in u: for small u, V^u percolates; for large u, V^u breaks apart. The model has connections to potential theory and to the Gaussian free field via isomorphism theorems and is used to study percolation phenomena in random media and to understand covering and hitting times of random walks.

Interlacement also appears in graph theory through the notion of interlacement graphs, particularly in circle graphs. In that context, two chords of a circle are interlaced if their endpoints alternate around the circle. The interlacement graph associated to a collection of chords has a vertex for each chord and edges between chords that interlace. This translation of geometric interweaving into a graph-theoretic structure helps analyze crossing patterns and chord arrangements.

Other uses occur in topology, knot theory, and physics, where the idea of interwoven or intertwined structures appears in descriptions of strands, filaments, or fibers.