Math behind everyday life: “black days”, their manifestation as traffic jams, and beyond

In our daily lives, we encounter numerous independent events, each occurring with varying probabilities over time. This work delves into the scientific background behind the inhomogeneous distribution of these events over time, often resulting in what we refer to as “black days”, where multiple events seem to converge at once. In the first part of the work we performed an analysis involving $D$ independent periodic and random sequences of events. Using the Uniform Manifold Approximation and Projection (UMAP) technique, we observed a clustering of event sequences on a two-dimensional manifold $M$ at a certain large $D$. We interpret this clustering as a signature of “black days”, which bears a clear resemblance to traffic jams in vehicle flow. In the second part of the work we examined in detail clustering patterns of independently distributed $N$ points within the corners of a $D$-dimensional cube when $1\ll N<D$. Our findings revealed that a transition to a single-component cluster occurs at a critical dimensionality, $D_{cr}$, via a nearly third-order phase transition. We demonstrate that for large $D$, the number of disjoint components exhibits a “saw-tooth” pattern as a function of $D$. Analyzing the spectral density, $\rho \left(\lambda \right)$, of the corresponding adjacency graph in the vicinity of the clustering transition we recover the singular “Lifshitz tail” behavior at the spectral boundary of $\rho \left(\lambda \right)$.