We study the noise sensitivity of the minimum spanning tree (MST) of the $n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $\varepsilon \gg n^{-1/3}$, then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if $\varepsilon \ll n^{-1/3}$, the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.

We survey some old and recent results concerning the Goldie dimension of modules and modular lattices and its properties which are counterparts of properties of the dimension of linear spaces.

In this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

A relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

A projective geometry of dimension (n - 1) can be defined as modular lattice with a spanning n-diamond of atoms (i.e.: n + 1 atoms in general position whose join is the unit of the lattice). The lemma we show is that one could equivalently define a projective geometry as a modular lattice with a spanning n-diamond that is (a) is generated (qua lattice) by this n-diamond and a coordinating diagonal and (b) every non-zero member of this coordinatizing diagonal is invertible. The lemma is applied to describe certain freely generated modular and Arguesian lattices.