Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes

G. Olshanski

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVIII, Tome 378 (2010), pp. 81-110

Résumé

The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function with the same index. Thus, the Laguerre symmetric functions form a two-parameter family of inhomogeneous bases in the algebra of symmetric functions. These new symmetric functions are obtained from the $N$-variate symmetric polynomials of the same name by a procedure of analytic continuation. The Laguerre symmetric functions are eigenvectors of a second order differential operator, which depends on the same two parameters and serves as the infinitesimal generator of an infinite-dimensional diffusion process $X(t)$. The process $X(t)$ admits approximation by some jump processes related to one more new family of symmetric functions, the Meixner symmetric functions. In equilibrium, the process $X(t)$ can be interpreted as a time-dependent point process on the punctured real line $\mathbb R\setminus\{0\}$, and the point configurations may be interpreted as doubly infinite collections of particles of two opposite charges and with log-gas-type interaction. The dynamical correlation functions of the equilibrium process have determinantal form: they are given by minors of the so-called extended Whittaker kernel, introduced earlier in a paper by Borodin and the author. Bibl. 28 titles.