This paper studies a random growth model in two dimensions related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converge in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE). The model is analyzed using probabilistic interpretations, including a randomly growing Young diagram, a totally asymmetric exclusion process, and a directed first-passage percolation model. The key findings include the convergence of shape fluctuations to the Tracy-Widom distribution and the computation of the asymptotic distribution of the appropriately rescaled random variable. The paper also discusses the connection between the model and random matrices, as well as the large deviation properties of the model. The results are supported by rigorous mathematical proofs and comparisons with known results in random matrix theory and combinatorics.This paper studies a random growth model in two dimensions related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converge in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE). The model is analyzed using probabilistic interpretations, including a randomly growing Young diagram, a totally asymmetric exclusion process, and a directed first-passage percolation model. The key findings include the convergence of shape fluctuations to the Tracy-Widom distribution and the computation of the asymptotic distribution of the appropriately rescaled random variable. The paper also discusses the connection between the model and random matrices, as well as the large deviation properties of the model. The results are supported by rigorous mathematical proofs and comparisons with known results in random matrix theory and combinatorics.