Activities This Week

The Coupon Collector’s Problem (CCP) reads as follows: how many drawings are needed on average in order to complete a collection of $n$ types of coupons, if at each step a single coupon is drawn uniformly randomly with replacement, independently of all the other drawings? This problem was introduced by De-Moivre over 300 years ago. We will discuss about a generalization of the problem, where instead of drawing a single coupon each time we draw a ``package” of coupons of size $s$ and ask the following question: how does the distribution over the collection of possible .packages affect the expected number of drawings needed to complete a collection?

Despite their potentially significant and dramatic consequences, large deviations in chaotic dynamics have been studied very little, with few existing theoretical results. We study large deviations of series of finite lengths N generated by chaotic maps. The distributions generally display an exponential decay with N, associated with large-deviation (rate) functions. We calculate the exact rate functions analytically for the doubling, tent, and logistic maps, and numerically for the cat map. In the latter case, we uncover a remarkable singularity of the rate function that we interpret as a second order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps. Our research lays the groundwork for the study of unusual trends of long duration in chaotic systems, such as heatwaves or droughts in climate models, or unusual mean growth rate of a pandemic over a long period of time. The talk is based on the recent work [N. R. Smith, Phys. Rev. E 106, L042202 (2022)].