Matrix formalism of excursion set theory: A new approach to statistics of dark matter halo counting

Excursion set theory (EST) is an analytical framework to study the large-scale structure of the Universe. EST introduces a procedure to calculate the number density of structures by relating the cosmological linear perturbation theory to the nonlinear structures in late time. In this work, we introduce a novel approach to reformulate the EST in matrix formalism. We propose that the matrix representation of EST will facilitate the calculations in this framework. The method is to discretize the two-dimensional plane of variance and density contrast of EST, where the trajectories for each point in the Universe lived there. The probability of having a density contrast in a chosen variance is represented by a probability ket. Naturally, the concept of the transition matrix pops up to define the trajectories. We also define the probability transition rate which is used to obtain the first up-crossing of trajectories and the number count of the structures. In this work we show that the discretization let us study the non-Markov processes by forcing them to look like a Wiener process. Also we discuss that the zero drift processes with Gaussian and also non-Gaussian initial conditions can be studied by this formalism. The continuous limit of the formalism is discussed, and the known Fokker-Planck dispersion equation is recovered. Finally we show that the probability of the most massive progenitors can be extracted in this framework.