## 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

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 \(1+\mathsf {Bin}(n-2,\frac{1}{n})\).Recall that all such probability spaces are isomorphic.

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

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

Note that the first step of the induction is satisfied trivially, as the height of

*T*is*L*.

## References

Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab.

**12**(54), 1454–1508 (2007)Alon, N.: The number of spanning trees in regular graphs. Random Struct. Algorithms

**1**(2), 175–181 (1990)Alon, N., Milman, V.D.: \(\lambda _1,\) isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Ser. B

**38**(1), 73–88 (1985)Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab.

**6**(23), 1–13 (2001)Bollobás, B., Borgs, C., Chayes, J., Riordan, O.: Percolation on dense graph sequences. Ann. Probab.

**38**(1), 150–183 (2010)Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Algorithms

**31**(1), 3–122 (2007)Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math.

**219**(6), 1801–1851 (2008)Frieze, A.: On the number of perfect matchings and Hamilton cycles in \(\epsilon \)-regular non-bipartite graphs. Electron. J. Combin.

**57**, 11 (2000)Grimmett, G.R.: Random labelled trees and their branching networks. J. Aust. Math. Soc. Ser. A

**30**(2), 229–237 (1980/81)Kirchhoff, G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72(12)

Kolchin, V.F.: Branching processes, random trees, and a generalized scheme of arrangements of particles. Math. Notes Acad. Sci. USSR

**21**(5), 386–394 (1977)Kostochka, A.V.: The number of spanning trees in graphs with a given degree sequence. Random Struct. Algorithms

**6**(2–3), 269–274 (1995)Lawler, G., Sokal, A.: Bounds on the \(L^2\) spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc.

**309**, 557–580 (1988)Levin, D., Peres, Y., with contributions by E. Wilmer:

*Markov Chains and Mixing Times*. American Mathematical Society, 2nd edn (2017). http://darkwing.uoregon.edu/dlevin/MARKOV/mcmt2e.pdfLovász, L.: Large Networks and Graph Limits. American Mathematical Society, Providence, RI (2012)

Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B

**96**(6), 933–957 (2006)Lyons, R.: Asymptotic enumeration of spanning trees. Combin. Prob. Comput.

**14**(4), 491–522 (2005)Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42. Cambridge University Press, New York (2016)

Matoušek, J., Nešetřil, J.: Invitation to Discrete Mathematics, 2nd edn. Oxford University Press, Oxford (2009)

Sinclair, A., Jerrum, M.: Approximate counting, uniform generation and rapidly mixing Markov chains. Inf. Comput.

**82**, 93–133 (1989)

## 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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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

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DOI: https://doi.org/10.1007/s10955-017-1933-5