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Excursion set theory (EST) is an analytical framework to study thelarge-scale structure of the Universe. EST introduces a procedure to calculatethe number density of structures by relating the cosmological linearperturbation theory to the nonlinear structures in late time. In this work, weintroduce a novel approach to reformulate the EST in matrix formalism. Wepropose that the matrix representation of EST will facilitate the calculationsin this framework. The method is to discretize the two-dimensional plane ofvariance and density contrast of EST, where the trajectories for each point inthe Universe lived there. The probability of having a density contrast in achosen variance is represented by a probability ket. Naturally, the concept ofthe transition matrix pops up to define the trajectories. We also define theprobability transition rate which is used to obtain the first up-crossing oftrajectories and the number count of the structures. In this work we show thatthe discretization let us study the non-Markov processes by forcing them tolook like a Wiener process. Also we discuss that the zero drift processes withGaussian and also non-Gaussian initial conditions can be studied by thisformalism. The continuous limit of the formalism is discussed, and the knownFokker-Planck dispersion equation is recovered. Finally we show that theprobability of the most massive progenitors can be extracted in this framework.

Matrix Formalism of Excursion Set Theory: A new approach to statistics of dark matter halo counting