TY - JOUR

T1 - The Local Limit of the Uniform Spanning Tree on Dense Graphs

AU - Hladký, Jan

AU - Nachmias, Asaf

AU - Tran, Tuan

N1 - Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - 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.

AB - 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.

KW - Benjamini-Schramm convergence

KW - Branching process

KW - Graph limits

KW - Graphon

KW - Uniform spanning tree

UR - http://www.scopus.com/inward/record.url?scp=85040344695&partnerID=8YFLogxK

U2 - 10.1007/s10955-017-1933-5

DO - 10.1007/s10955-017-1933-5

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AN - SCOPUS:85040344695

SN - 0022-4715

VL - 173

SP - 502

EP - 545

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3-4

ER -