Effect of microscopic pausing time distributions on the dynamical limit shapes for random Young diagrams

The irreducible decomposition of successive restriction and induction of irreducible representations of a symmetric group gives rise to a Markov chain on Young diagrams keeping the Plancherel measure invariant. Starting from this Res-Ind chain, we introduce a not necessarily Markovian continuous time random walk on Young diagrams by considering a general pausing time distribution between jumps according to the transition probability of the Res-Ind chain. We show that, under appropriate assumptions for the pausing time distribution, a diffusive scaling limit brings us concentration at a certain limit shape depending on macroscopic time which leads to a similar consequence to the exponentially distributed case studied in our earlier work. The time evolution of the limit shape is well described by using free probability theory. On the other hand, we illustrate an anomalous phenomenon observed with a pausing time obeying a one-sided stable distribution, heavy-tailed without the mean, in which a nontrivial behavior appears under a non-diffusive regime of the scaling limit.Comment: 25 pages, 2 figures, Introduction modified with results unchanged, A reference adde

Teaching Marginal Analysis: On the importance of emphasising the second-order condition

Students employ marginal analysis in investigating various problems regarding resource allocation. The majority of first-year students, however, seem to struggle in applying it to a firm's profit maximisation problem. Their confusion appears to stem from textbooks for introductory microeconomics courses oversimplifying some subtle points regarding the profit maximisation problem. This paper (1) points out how oversimplification may impede students' understanding, contrary to the instructors' pedagogical intent, and (2) suggests a complementary method to enhance their understanding of marginal analysis in the course of teaching a firm's profit maximisation problem.

The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.Comment: 9 page