The emergence of new phenomena from the interplay between randomness and geometry is a vital mechanism in many instances in physics, such as structure formation, percolation, phase transitions, as well as the correlated spatial randomness seen in fluids, glasses and jammed packings. However, rigorous results are often rare in models of interest in physics, while many rigorous results from mathematics are often little appreciated in the physics contexts. From a physics perspective, simple exactly solvable models allow one to elucidate the underlying mechanisms of phenomena with great clarity.