2021/07/28 by Yevgeniy Kovchegov, Kovchegov, Yevgeniy, Peter T. Otto +1
Engineering · #05C50 #47J07 #60C05 #60J90 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Sports Dynamics and Biomechanics
paper · pdf · doi:10.48550/arxiv.2107.13162
openalex publication_date 2021/07/28 · openalex created_date 2023/11/25 · openalex updated_date 2026/07/28
In this paper we show the existence of the minimal solution to the multidimensional Lambert-Euler inversion, a multidimensional generalization of [-e-1 ,0) branch of Lambert W function W0(x). Specifically, for a given nonnegative irreducible symmetric matrix V ∈ ℝk × k, we show that for \bf u∈(0,∞)k, if equation yj exp\-\bf ejT V \bf y \ = uj ~~~~~~∀ j=1,...,k, has at least one solution, it must have a minimal solution \bf y^*, where the minimum is achieved in all coordinates yj simultaneously. Moreover, such \bf y^* is the unique solution satisfying ρ(V D[y^*j] ) ≤ 1, where D[y^*j]=\sf diag(yj^*) is the diagonal matrix with entries y^*j and ρ denotes the spectral radius. Our main application is in the vector-multiplicative coalescent process. It is a coalescent process with k types of particles and vector-valued weights that begins with α1n+...+αk n particles partitioned into types of respective sizes, and in which two clusters of weights \bf x and \bf y would merge with rate (\bf x\sf T V \bf y)/n. We use combinatorics to solve the corresponding modified Smoluchowski equations, obtained as a hydrodynamic limit of vector-multiplicative coalescent as n → ∞, and use multidimensional Lambert-Euler inversion to establish gelation and find a closed form expression for the gelation time. We also find the asymptotic length of the minimal spanning tree for a broad range of graphs equipped with random edge lengths.