Multidimensional Lambert-Euler inversion and vector-multiplicative coalescent processes

Abstract

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 $W_0(x)$. Specifically, for a given nonnegative irreducible symmetric matrix $V\in {\mathbb{R}}^{k\times k}$, we show that for $\mathbf{u}\in (0,\infty {)}^k$, if equation $$y_j\exp \{-{\mathbf{e}}_j^{T}V\mathbf{y}\}=u_j\forall j=1,...,k,$$ has at least one solution, it must have a minimal solution ${\mathbf{y}}^{\ast }$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such ${\mathbf{y}}^{\ast }$ is the unique solution satisfying $\rho \left(VD[y_j^{\ast }]\right)\le 1$, where $D[y_j^{\ast }]=\mathsf{diag}(y_j^{\ast })$ is the diagonal matrix with entries $y_j^{\ast }$ and $\rho$ 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 ${\alpha }_{1}n+...+{\alpha }_{k}n$ particles partitioned into types of respective sizes, and in which two clusters of weights $\mathbf{x}$ and $\mathbf{y}$ would merge with rate $({\mathbf{x}}^{\mathsf{T}}V\mathbf{y})/n$. We use combinatorics to solve the corresponding modified Smoluchowski equations, obtained as a hydrodynamic limit of vector-multiplicative coalescent as $n\to \infty$, 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.

28 pages