From correlated random walks to the bivariate Gaussian: a computational approach to covariance and principal fluctuation directions

We present a pedagogical framework for introducing correlations in stochastic systems through a minimal model of two random walkers moving on parallel one-dimensional lattices. Correlations are imposed microscopically on the simultaneous increments: at each step, the walkers move in the same direction with probability (1+c)/2 and in opposite directions with probability (1-c)/2, where -1≤c≤1. This construction preserves the symmetric and unbiased marginal motion of each walker while making c exactly equal to the correlation coefficient of their increments. We derive the covariance matrix of the walkers' positions and show that the same correlation coefficient is preserved for any number of steps. For |c|<1, the multivariate central limit theorem leads to an asymptotic bivariate Gaussian distribution. The relation between the covariance matrix, its inverse, and the quadratic form in the Gaussian exponent is developed explicitly, providing a concrete interpretation of the cross term and of the principal directions of the distribution. Collective and relative coordinates diagonalize the covariance matrix and lead to the asymptotic mean absolute separation <|x₁-x₂|>≃2ℓ√(((N(1-c))/π)). Monte Carlo simulations confirm the predicted variances, covariance, joint Gaussian distribution, and relative-separation statistics. The model provides an accessible route from elementary random walks to multivariate Gaussian statistics, covariance matrices, quadratic forms, and principal fluctuation directions. It is intended for undergraduate statistical mechanics courses and may be used as a classroom example, a guided computational activity, or a short student project.