Benjamini-Schramm convergence and subtrees of trees
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918489407619072 |
|---|---|
| author | Cambie, Stijn Wagner, Stephan Wang, Ruoyu |
| author_facet | Cambie, Stijn Wagner, Stephan Wang, Ruoyu |
| contents | In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval $[0,1]$ for both general trees and series-reduced trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07664 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Benjamini-Schramm convergence and subtrees of trees Cambie, Stijn Wagner, Stephan Wang, Ruoyu Combinatorics In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval $[0,1]$ for both general trees and series-reduced trees. |
| title | Benjamini-Schramm convergence and subtrees of trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.07664 |