The critical tree of a renormalization model as a growth-fragmentation process

Overview & Synopsis

We study a branching system which describes the evolution indexed by a continuous time parameter ranging in [0, 1) of a population of cells; the size of each cell increases deterministically and linearly except when the cell splits into two daughter cells. The system appears as the scaling limit of the critical tree in the family of hierarchical renormalization models studied in [12], conditioned on survival; it is also a growth-fragmentation process in the sense of Bertoin [3]. We are interested in the empirical measure of the process representing the sizes of the cells that are alive at time t ∈ [0, 1), and establish a general result, called the master formula, for exponential functionals of the empirical measure. The formula allows to determine the joint distribution of the sum of cell sizes and the number of cells at time t, which improves a previous result by Hu, Mallein and Pain [17] who proved joint weak convergence of these two quantities when t → 1 −. The main result in our paper, established also relying on the master formula, is a law of large numbers for the empirical measure when t → 1 − , the limiting distribution explicitly identified. Our system can be viewed as an exactly solvable example of a growth-fragmentation process.