Editing Random minimum spanning tree (section)

In mathematics, a '''random minimum spanning tree''' may be formed by assigning [[Independence (probability theory)|independent]] random weights from some distribution to the edges of an [[undirected graph]], and then constructing the [[minimum spanning tree]] of the graph.

[[File:Random minimum spanning tree.svg|thumb|380px|Random minimum spanning tree on the same graph but with randomized weights.]]

When the given graph is a [[complete graph]] on {{mvar|n}} vertices, and the edge weights have a continuous [[Cumulative distribution function|distribution function]] whose derivative at zero is {{math|''D'' > 0}}, then the expected weight of its random minimum spanning trees is bounded by a constant, rather than growing as a function of {{mvar|n}}. More precisely, this constant tends in the limit (as {{mvar|n}} goes to infinity) to {{math|''ζ''(3)/''D''}}, where {{mvar|ζ}} is the [[Riemann zeta function]] and {{math|''ζ''(3) ≈ 1.202}} is [[Apéry's constant]]. For instance, for edge weights that are uniformly distributed on the [[unit interval]], the derivative is {{math|1=''D'' = 1}}, and the limit is just {{math|''ζ''(3)}}.{{r|frieze}} For other graphs, the expected weight of the random minimum spanning tree can be calculated as an integral involving the [[Tutte polynomial]] of the graph.{{r|steele}}

In contrast to [[uniform spanning tree|uniformly random spanning trees]] of complete graphs, for which the typical [[Diameter (graph theory)|diameter]] is proportional to the square root of the number of vertices, random minimum spanning trees of complete graphs have typical diameter proportional to the cube root.{{r|goldschmidt|abgm}}

Random minimum spanning trees of [[grid graph]]s may be used for [[invasion percolation]] models of liquid flow through a porous medium,{{r|d3m3h}} and for [[maze generation]].{{r|foltin}}