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.
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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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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