Recent zbMATH articles in MSC 05C81https://zbmath.org/atom/cc/05C812021-06-15T18:09:00+00:00WerkzeugThe diameter of uniform spanning trees in high dimensions.https://zbmath.org/1460.050572021-06-15T18:09:00+00:00"Michaeli, Peleg"https://zbmath.org/authors/?q=ai:michaeli.peleg"Nachmias, Asaf"https://zbmath.org/authors/?q=ai:nachmias.asaf"Shalev, Matan"https://zbmath.org/authors/?q=ai:shalev.matanSummary: We show that the diameter of a uniformly drawn spanning tree of a connected graph on \(n\) vertices which satisfies certain high-dimensionality conditions typically grows like \(\Theta (\sqrt{n})\). In particular this result applies to expanders, finite tori \(\mathbb{Z}_m^d\) of dimension \(d\ge 5\), the hypercube \(\{0,1\}^m\), and small perturbations thereof.Loop-erased random walk as a spin system observable.https://zbmath.org/1460.051792021-06-15T18:09:00+00:00"Helmuth, Tyler"https://zbmath.org/authors/?q=ai:helmuth.tyler"Shapira, Assaf"https://zbmath.org/authors/?q=ai:shapira.assafSummary: The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.Polynomial approximation for the number of all possible endpoints of a random walk on a metric graph.https://zbmath.org/1460.051782021-06-15T18:09:00+00:00"Chernyshev, Vsevolod"https://zbmath.org/authors/?q=ai:chernyshev.vsevolod-l"Tolchennikov, Anton"https://zbmath.org/authors/?q=ai:tolchennikov.anton-aSummary: The asymptotics of the number of possible endpoints of a random walk on a metric graph with incommensurable edge lengths is found.
For the entire collection see [Zbl 1409.68021].