We will discuss a toy model of heavy tails and show how this does not follow central limit behavior. We will then see how this relates to models in physics including random matrices. In the random matrix setting, we equate limiting spectral distributions (LSD) to spectral measures of rooted graphs. The LSD result also includes matrices with i.i.d. entries (up to self-adjointness) having infinite second moments, but following central limit behavior. In this case, the rooted graph is the natural numbers rooted at one, so the LSD is well-known to be the semi-circle law.