# The Local Limit of the Uniform Spanning Tree on Dense Graphs

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## Abstract

Let G be a connected graph in which almost all vertices have linear degrees and let $$\mathcal {T}$$ 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 $$\mathcal {T}$$ is isomorphic to F. We deduce from this that if $$\{G_n\}$$ is a sequence of such graphs converging to a graphon W, then the uniform spanning tree of $$G_n$$ 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 $$e^{-1}-\mathsf {o}(1)$$, the density of vertices of degree 2 is at most $$e^{-1}+\mathsf {o}(1)$$ and the density of vertices of degree $$k\geqslant 3$$ is at most $${(k-2)^{k-2} \over (k-1)! e^{k-2}} + \mathsf {o}(1)$$. These bounds are sharp.

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## Notes

1. Prüfer codes, see e.g. [19, p. 245], provide a standard bijection between spanning trees of $$K_n$$ and and words of length $$n-2$$ over the alphabet $$V(K_n)$$. A quick look at this bijection shows the number of occurrences of any letter $$v\in V(K_n)$$ in a Prüfer code is the degree of the vertex v in the corresponding spanning tree decreased by 1. Therefore, the degree of v in a uniform spanning tree has indeed distribution $$1+\mathsf {Bin}(n-2,\frac{1}{n})$$.

2. Recall that all such probability spaces are isomorphic.

3. Such a sequence can be obtained for example by taking typical inhomogeneous random graphs $$\mathbb G(n,W^+)$$, see [15, Lemma 10.16].

4. Note that later, in the proof of Lemma 3.10, we shall be forced to use larger implicit constants in (b) of Definition 2.9.

5. Strictly speaking, when some but not all coordinates $$r_i$$ are 0 or h, these are not cubes but rectangular prisms. This is however not important.

6. Note that in this case, the cut-norm really has to be defined over all rectangles in (6), and not only over all squares.

7. At this moment, we have not established Theorem 1.3 nor Theorem 1.1. However, the fact $$\mathrm {Freq}(T;W)\leqslant 1$$ did not rely on the validity of either of these theorems, but rather followed from making the connection to the branching process $$\kappa _W$$.

8. Note that the first step of the induction is satisfied trivially, as the height of T is L.

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## Acknowledgements

Hladký and Tran were supported by the Czech Science Foundation, Grant Number GJ16-07822Y and by institutional support RVO:67985807. Hladký is supported by the Alexander von Humboldt Foundation. Nachmias is supported by ISF Grant 1207/15, and ERC starting Grant 676970 RANDGEOM. Part of this work was done while Hladký and Tran were visiting Tel Aviv University. They thank the department of mathematical sciences at TAU for their hospitality. The trip was funded by the Czech Academy of Sciences. We thank Omer Angel and Tom Hutchcroft for their assistance with the proof of Lemma 3.2 and to Michael Krivelevich and Wojciech Samotij for useful discussions.

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Correspondence to Asaf Nachmias.

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Hladký, J., Nachmias, A. & Tran, T. The Local Limit of the Uniform Spanning Tree on Dense Graphs. J Stat Phys 173, 502–545 (2018). https://doi.org/10.1007/s10955-017-1933-5