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 ∈ℝ^k × k and a vector u∈ (0,∞ )^k , we show that, if the system of equations y_j exp{-e_j^ V y} = u_j ∀ j=1,… ,k, has at least one solution, it must have a minimal solution y^* , where the minimum is achieved in all coordinates y_j simultaneously. Moreover, such y^* is the unique solution satisfying ρ( V D[y^*_j] ) ≤ 1 , where D[y^*_j]=(y_j^*) is the diagonal matrix with entries y^*_j and ρ 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(x, y)=x^ V 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.