Abstract
We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of mathbb {Z}^d, dge 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of mathbb {Z}^d, dge 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.
Highlights
2.1 Loop-erased random walk and Wilson’s algorithmLet G be a network
If G is transient, the sequence of laws of these random oriented spanning trees converge weakly to the law of a random oriented spanning forest of G, which is known as the oriented wired uniform spanning forest, and from which we can recover the usual WUSF by forgetting the orientation. (This assertion follows from the proof of [20, Theorem 5.1].) It is seen that the oriented wired uniform spanning forest of G is almost surely an oriented essential spanning forest of G, that is, an oriented spanning forest of G such that every vertex of G has exactly one oriented edge emanating from it in the forest
For every n ≥ 0 and every pair of vertices x, y in G with d(x, y) ≤ n. It follows from the work of Hebisch and Saloff-Coste [30] that every transitive graph of polynomial volume growth satisfies Gaussian heat-kernel estimates, as does every bounded degree network with edge conductances bounded between two positive constants that is roughisometric to a transitive graph of polynomial growth
Summary
When G is a regular tree of degree k ≥ 3, the components of the WUSF are distributed exactly as augmented critical binomial Galton–Watson trees conditioned to survive forever, and in this case all of our results are classical [13,43,54]. Peres and Revelle [62] proved that the USTs of large d-dimensional tori converge under rescaling (with respect to the Gromov-weak topology) to Aldous’s continuum random tree when d ≥ 5. They proved that their result extends to other sequences of finite transitive graphs satisfying a heat-kernel upper bound similar to the one we assume here. In forthcoming work with Sousi, we build upon the methods of this paper to analyze related problems concerning the uniform spanning tree in Z3 and Z4
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