Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
\textit{R. Pemantle} [Ann. Probab. 19, No. 4, 1559--1574 (1991; Zbl 0758.60010)] showed that the \(n \to \infty\) local limit of the uniform spanning tree on \([-n,n]^2 \cap \mathbb{Z}^2\) (with nearest-neighbour edges) defines a uniform spanning tree (UST) \(\mathcal{U}\) on \(\mathbb{Z}^2\) which is a.s.~a spanning tree, i.e., any two vertices of \(\mathbb{Z}^2\) are connected by a unique path in \(\mathcal{U}\). Denote by \(d_\mathcal{U}\) the intrinsic (shortest-path) metric on the graph \(\mathcal{U}\), and by \(\mu_\mathcal{U}\) the uniform measure that places unit mass at each vertex of \(\mathcal{U}\). Extend the metric space \((\mathcal{U},d_\mathcal{U})\) to a complete and locally compact real tree by adding unit line segments along the edges of \(\mathcal{U}\), and view the measure \(\mu_\mathcal{U}\) as a locally finite Borel measure on the real tree \((\mathcal{U},d_\mathcal{U})\). Moreover, embed this real tree in \(\mathbb{R}^2\) as a spatial tree via the continuous map \(\phi_\mathcal{U} : \mathcal{U} \to \mathbb{R}^2\) which is the identity on the vertices of \(\mathcal{U}\), with linear interpolation along edges. Finally, introduce a root at \(0 \in \mathbb{Z}^2\). The random quintuple \((\mathcal{U},d_\mathcal{U}, \mu_\mathcal{U}, \phi_\mathcal{U},0)\) is the object of the first main result of the paper, which states that its law is tight under rescaling in the appropriate space of measured, rooted spatial trees. Namely, it is shown that the laws \(\mathbf{P}_\delta\), \(\delta \in (0,1)\), associated with \((\mathcal{U}, \delta^\kappa d_\mathcal{U}, \delta^2 \mu_\mathcal{U}, \delta \phi_\mathcal{U}, 0)\) are tight. Here, \(\kappa = 5/4\) is the growth exponent of the loop-erased random walk on \(\mathbb{Z}^2\). The topology preserves information about the intrinsic metric \(d_\mathcal{U}\) and measure \(\mu_\mathcal{U}\), and hence strengthens earlier scaling results of \textit{M. Aizenman} et al. [Random Struct. Algorithms 15, No. 3--4, 319--367 (1999; Zbl 0939.60031)] and \textit{O. Schramm} [Isr. J. Math. 118, 221--288 (2000; Zbl 0968.60093)]. In addition, various properties of the subsequential limits of \((\mathbf{P}_\delta)_{\delta \in (0,1)}\) are obtained, including results on the intrinsic metrics, measures, and the emdeddings; in particular, it is shown that a.s.~the real tree limit has Hausdorff dimension \(2/\kappa = 8/5\) and has precisely one end at infinity. The second topic of the paper is the scaling limits of the simple random walk (SRW) on the UST \(\mathcal{U}\). It is shown that the annealed law of the SRW (averaged over the environment \(\mathcal{U}\)) is tight under a suitable rescaling. This result is deduced from a general theorem on convergence of SRW on graph trees, which extends earlier work of Croydon. For the limiting processes, which are diffusions on random real trees embedded in Euclidean space, detailed transition density estimates are derived.
scientific article