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In this paper we show the existence of the minimal solution to themultidimensional Lambert-Euler inversion, a multidimensional generalization of$[-e^{-1} ,0)$ branch of Lambert W function $W_0(x)$. Specifically, for a givennonnegative irreducible symmetric matrix $V \in \mathbb{R}^{k \times k}$, weshow 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 musthave a minimal solution ${\bf y}^*$, where the minimum is achieved in allcoordinates $y_j$ simultaneously. Moreover, such ${\bf y}^*$ is the uniquesolution satisfying $\rho\left(V D[y^*_j] \right) \leq 1$, where $D[y^*_j]={\sfdiag}(y_j^*)$ is the diagonal matrix with entries $y^*_j$ and $\rho$ denotesthe spectral radius.  Our main application is in the vector-multiplicative coalescent process. Itis a coalescent process with $k$ types of particles and vector-valued weightsthat begins with $\alpha_1n+...+\alpha_k n$ particles partitioned into types ofrespective 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 tosolve the corresponding modified Smoluchowski equations, obtained as ahydrodynamic limit of vector-multiplicative coalescent as $n \to \infty$, anduse multidimensional Lambert-Euler inversion to establish gelation and find aclosed form expression for the gelation time.  We also find the asymptotic length of the minimal spanning tree for a broadrange of graphs equipped with random edge lengths.

Multidimensional Lambert-Euler inversion and vector-multiplicative coalescent processes