UC Berkeley

This thesis consists of three projects related to the asymptotic behavior of large combinatorial structures. In the first project, we bound the potential energy for a collection of charged particles on the positive real line with logarithmic interaction potentials. This was conjectured by Pohst in 1977 with a pure number-theoretic motivation; it arises when describing bounds between the discriminant and regulator of totally real number fields. The proof is purely combinatorial and involves a careful description of the possible sign configurations of the particle system. In the second project, we study the global fluctuation of random standard Young tableaux. We introduce a Laplace transform for partitions, with which we characterize a law of large numbers, a central limit theorem for random partitions and a multilevel central limit theorem. We consider the fluctuations of random surfaces associated with: the Plancherel growth process, distributions induced by extreme characters of the infinite symmetric group and random standard Young tableaux with fixed shape. We show that all converge to a conditioned Gaussian Free Field. We also describe the fluctuations of sublinear random standard Young tableaux, prove a conjecture of Pittel and Romik from 2004 and obtain several combinatorial results of independent interest. The proofs are based on the study of certain operators on the Gelfand--Tsetlin algebra of the symmetric group. In the third project, we introduce random matrix models that are heavily motivated by the Plancherel growth model. In the first model, we construct a non-commutative coupling between various random Plancherel distributed partitions induced by the regular character on the infinite symmetric group. In parallel, we propose a traceless Wigner random matrix ensemble and we show that both constructions have global fluctuations described by collections of correlated Gaussian Free Fields. In the second model, we define a trace-normalized random matrix ensemble which induces a new higher-conditioned Gaussian Free Field. The proofs are based on careful applications of the moment method.