---
tag: ET73YH2T29
kind: concept
name: Loop-erased random walk
msc: [60G50, 82B41, 05C81]
sources:
  - url: https://en.wikipedia.org/wiki/Loop-erased_random_walk
same_as:
  wikidata: Q6675770
  wikipedia: Loop-erased random walk
---

## Summary

A loop-erased random walk is the simple path obtained from a [[random walk|04VZE3FZXF]] by erasing its loops in the order in which they are created.
It is a random self-avoiding path, but its law differs from the uniform [[self-avoiding walk|E2BVEB570S]].
Through Wilson's algorithm it is the building block of the uniform [[spanning tree|0S4ZEZM7KE]] of a graph.
In the plane its scaling limit is the Schramm–Loewner evolution with parameter $2$.

## Explanations

### L1 human

Let a random walker wander until it reaches a target, and record its path.
Each time the walker comes back to a place it has already visited, cut out the loop it has just made.
What remains is a path from start to target that never crosses itself.
For example, on the integer line the path $0, 1, 2, 1, 2, 3$ becomes $0, 1, 2, 3$ after the loop $2, 1, 2$ is erased.