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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 [[
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}}
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| last1 = Addario-Berry | first1 = Louigi
| last2 = Broutin | first2 = Nicolas
| last3 = Goldschmidt | first3 = Christina | author3-link = Christina Goldschmidt
| last4 = Miermont | first4 = Grégory | author4-link = Grégory Miermont
| doi = 10.1214/16-AOP1132
| issue = 5
| journal = [[Annals of Probability]]
| pages = 3075–3144
| title = The scaling limit of the minimum spanning tree of the complete graph
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| title = Computer Simulation Studies in Condensed-Matter Physics XVI: Proceedings of the Fifteenth Workshop, Athens, GA, USA, February 24–28, 2003
| volume = 95
| year = 2004
}}.</ref>
<ref name=foltin>{{citation|url=http://www.martinfoltin.sk/mazes/thesis.pdf|title=Automated Maze Generation and Human Interaction|first=Martin|last=Foltin|series=Diploma Thesis|publisher=Masaryk University, Faculty of Informatics|___location=Brno|year=2011}}.</ref>
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<ref name=steele>{{citation
| last = Steele | first = J. Michael | author-link = J. Michael Steele
| editor1-last = Chauvin | editor1-first = Brigitte
| editor2-last = Flajolet | editor2-first = Philippe | editor2-link = Philippe Flajolet
| editor3-last = Gardy | editor3-first = Danièle
| editor4-last = Mokkadem | editor4-first = Abdelkader
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| series = Trends in Mathematics
| title = Mathematics and Computer Science II: Algorithms, Trees, Combinatorics and Probabilities, Proceedings of the 2nd Colloquium, Versailles-St.-Quentin, France, September 16–19, 2002
| year = 2002| isbn = 978-3-0348-9475-3 }}</ref>
}}
{{Combin-stub}}▼
[[Category:Spanning tree]]
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