The Local Limit of the Uniform Spanning Tree on Dense Graphs

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

The Local Limit of the Uniform Spanning Tree on Dense Graphs

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ16-07822Y" target="_blank" >GJ16-07822Y: Extremální teorie grafů a aplikace</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Statistical Physics

ISSN

0022-4715

e-ISSN

—

Svazek periodika

173

Číslo periodika v rámci svazku

3-4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

44

Strana od-do

502-545

Kód UT WoS článku

000450490500004

EID výsledku v databázi Scopus

2-s2.0-85040344695