In 2014, Sourav Chatterjee demonstrated for a class of Gaussian models of disorder that three phenomena occur together if any one of them occurs. These effects are superconcentration, in which the fluctuation of a random variable is smaller than that present in the classical context of a central limit theorem; multiple valleys, in which many geometrically distinct rivals nearly attain an extreme statistic for the system; and chaos, in which the system is very sensitive to perturbation by random noise. We will discuss Levy's construction of Brownian motion, and a natural dynamical enhancement of Brownian motion involving Ornstein-Uhlenbeck processes; and we will see how these examples may be interpreted in Chatterjee's theory. We will review significant examples of chaotic behaviour when statistical mechanical systems are enhanced by random dynamics. These examples include such enhancements of critical percolation, ensembles of random matrices and last passage percolation.

Random tilings are an important area in probability, combinatorics, and statistical mechanics.

Unfortunately even enumeration of tilings of regions is typically NP-hard except in very special cases.

We propose a variant of the usual random tiling model, called the multinomial tiling model,
which is both very general and also tractable in the sense of allowing exact computation of growth rates.

We can also compute Gaussian scaling limits, find crystallization phenomena, and even random quasicrystals.