Probability distribution for coherent transport of random waves

Random-matrix theory has been remarkably successful in predicting universal statistical properties of wave transport in complex media by averaging over ensembles of random media. However, wavefront-shaping experiments typically probe a single fixed medium with varying input states, calling for a probability theory from the perspective of random waves rather than random media. Here, we establish such a theory. We prove that the probability distribution of power transmissivity for random coherent waves through any fixed medium is a fundamental B-spline with knots at the transmission eigenvalues. We analyze the distribution's shape, bounds, and moments and show that it converges to a Gaussian in the many-channel limit, determined solely by the mean and variance of the eigenvalues. We derive bounds on tail probabilities that quantify the experimental accessibility of open channels. Our framework extends to other transport observables, including reflectivity and absorptivity, and is validated through full-wave simulations of disordered waveguides.