The Local Limit of the Uniform Spanning Tree on Dense Graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F18%3A00502247" target="_blank" >RIV/67985807:_____/18:00502247 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s10955-017-1933-5" target="_blank" >http://dx.doi.org/10.1007/s10955-017-1933-5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10955-017-1933-5" target="_blank" >10.1007/s10955-017-1933-5</a>

Result language
angličtina
Original language name
The Local Limit of the Uniform Spanning Tree on Dense Graphs
Original language description
Let G be a connected graph in which almost all vertices have linear degrees and let T be a uniform spanning tree of G. For any fixed rooted tree F of height r we compute the asymptotic density of vertices v for which the r-ball around v in T is isomorphic to F. We deduce from this that if {Gn} is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of Gn locally converges to a multi-type branching process defined in terms of W. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least e^-1 - o(1), the density of vertices of degree 2 is at most e^-1 + o(1) and the density of vertices of degree k⩾ 3 is at most (k-2)k-2(k-1)!ek-2+o(1). These bounds are sharp.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ16-07822Y" target="_blank" >GJ16-07822Y: Extremal graph theory and applications</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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