Multidimensional Lambert-Euler inversion and vector-multiplicative coalescent processes

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 ${\bf u}\in(0,\infty)^k$, if equation $$y_j \exp\{-{\bf e}_j^T V {\bf y} \} = u_j ~~~~~~\forall j=1,...,k,$$ has at least one solution, it must have a minimal solution ${\bf y}^*$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such ${\bf y}^*$ is the unique solution satisfying $\rho\left(V D[y^*_j] \right) \leq 1$, where $D[y^*_j]={\sf diag}(y_j^*)$ is the diagonal matrix with entries $y^*_j$ 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 $\alpha_1n+...+\alpha_k n$ 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 $n \to \infty$, 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.