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In a series of papers [22-24] by Bufetov and Gorin, Schur generatingfunctions as the Fourier transforms on the unitary group $U(N)$, are introducedto study the asymptotic behaviors of random $N$-particle systems. We introduceand study the Jack generating functions of random $N$-particle systems. Inspecial cases, this can be viewed as the Fourier transforms on the Gelfandpairs $(GL_N(\mathbb R), O(N))$, $(GL_N(\mathbb C), U(N))$ and $(GL_N(\mathbbH), Sp(N))$. Our main results state that the law of large numbers and thecentral limit theorems for such particle systems, is equivalent to certainconditions on the germ at unity of their Jack generating functions. Our maintool is the Nazarov-Sklyanin operators [50], which have Jack symmetricfunctions as their eigenfunctions. As applications, we derive asymptotics ofJack characters, prove law of large numbers and central limit theorems for theLittlewood-Richardson coefficients of zonal polynomials, and show that thefluctuations of the height functions of a general family of nonintersectingrandom walks are asymptotically equal to those of the pullback of the Gaussianfree field on the upper half plane.

Law of Large Numbers and Central Limit Theorems by Jack Generating Functions