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 $V \in \mathbb {R}^{k \times k}$ and a vector $\textbf{u}\in (0,\infty )^k$, we show that, if the system of equations

$$\begin{aligned} y_j \exp \big \{-\textbf{e}_j^{\textsf {T}} V \textbf{y} \big \} = u_j \qquad \forall j=1,\ldots ,k, \end{aligned}$$

has at least one solution, it must have a minimal solution $\textbf{y}^*$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such $\textbf{y}^*$ is the unique solution satisfying $\rho \left( V D[y^*_j] \right) \le 1$, where $D[y^*_j]=\textsf {diag}(y_j^*)$ is the diagonal matrix with entries $y^*_j$ and $\rho$ denotes the spectral radius. Our main application is in the analysis of the vector-multiplicative coalescent process. It is a coalescent process with k types of particles and k-dimensional vector-valued cluster weights representing the composition of a cluster by particle types. The clusters merge according to the vector-multiplicative kernel $K(\textbf{x}, \textbf{y})=\textbf{x}^{\textsf {T}} V \textbf{y}$. First, we derive some new combinatorial results, and use them to solve the corresponding modified Smoluchowski equations obtained as a hydrodynamic limit of vector-multiplicative coalescent. Then, we 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.