A loop-erased random walk is the self-avoiding path obtained by chronologically erasing loops from a random-walk path.
Chronological loop erasure removes each loop from a finite or transient infinite path as soon as the loop is formed, producing a self-avoiding path.
On a finite undirected graph, the reversal of a loop-erased random walk from to has the law of a loop-erased random walk from to . The uniform-spanning-tree path representation proves the identity.
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Loop-erased random walk (LERW) is a mathematical construct and a type of random walk that is particularly interesting in the fields of probability theory and statistical mechanics. It can be thought of as a model for exploring the behavior of paths in a random environment while avoiding certain obstacles (loops). Here's how it works: 1. **Random Walk**: Begin with a simple random walk on a lattice (for example, the integer grid in two dimensions), starting from an origin point.