Georgia Institute of Technology College of Sciences

Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs.  In work in progress with Robert Chang (Rhodes College), we introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with positive Hurst index s -1/2.  For a special Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams.  The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp.  The global Gaussian fluctuations depend on s and, if we continue s to negative values, coincides with the process in Kerov's CLT for s = - 1/2.  Although it might be possible to give a direct explanation for this coincidence by regularization, in this talk we give an indirect dynamical explanation by combining (i) results of Eliashberg and Dubrovin for a specific Hamiltonian QFT and (ii) the fact that in Hamiltonian systems, at short time scales, the quantum evolution of pure Gaussian wavepacket initial data agrees statistically with the classical evolution of mixed Gaussian random initial data.