Fluctuations of particle systems determined by Schur generating functions

Abstract

We develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems. We introduce and study the notion of the Schur generating function of a random discrete configuration. Our main result provides a Central Limit Theorem (CLT) for such a configuration given certain conditions on the Schur generating function. As applications of this approach, we prove CLT's for several probabilistic models coming from asymptotic representation theory and statistical physics, including random lozenge and domino tilings, non-intersecting random walks, decompositions of tensor products of representations of unitary groups.

arXiv comment65 pages, 4 figures. v3: some misprints corrected compared to the journal version

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Cited by in corpus (19)

• Fourier transform on high-dimensional unitary groups with applications to random tilings
• Bulk universality for random lozenge tilings near straight boundaries and for tensor products
• A periodic hexagon tiling model and non-Hermitian orthogonal polynomials
• Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q,t)$-Gelfand-Tsetlin Graph
• Fluctuations of interlacing sequences
• Matrix addition and the Dunkl transform at high temperature
• Schur function at generic points and limit shape of perfect matchings on contracting square hexagon lattices with piecewise boundary conditions
• Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks
• Lozenge tilings and the Gaussian free field on a cylinder
• Airy Point Process via Supersymmetric Lifts
• Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions
• Log-gases on a quadratic lattice via discrete loop equations and q-boxed plane partitions
• Fluctuations in the Aztec diamonds via a space-like maximal surface in Minkowski 3-space
• A Quantized Analogue of the Markov-Krein Correspondence
• $\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems
• How to Draw a Correlation Function
• Asymptotics of Schur functions on almost staircase partitions
• A power series identity and Bessel-type integrals over unitary groups