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Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes. (arXiv:1804.07994v3 [math.PR] UPDATED)
来源于:arXiv
Rosengren and Schlosser introduced notions of ${\it R}_N$-theta functions for
the seven types of irreducible reduced affine root systems, ${\it R}_N={\it
A}_{N-1}$, ${\it B}_{N}$, ${\it B}^{\vee}_N$, ${\it C}_{N}$, ${\it
C}^{\vee}_N$, ${\it BC}_{N}$, ${\it D}_{N}$, $N \in \mathbb{N}$, and gave the
Macdonald denominator formulas. We prove that, if the variables of the ${\it
R}_N$-theta functions are properly scaled with $N$, they construct seven sets
of biorthogonal functions, each of which has a continuous parameter $t \in (0,
t_{\ast})$ with given $0< t_{\ast} < \infty$. Following the standard method in
random matrix theory, we introduce seven types of one-parameter ($t \in (0,
t_{\ast})$) families of determinantal point processes in one dimension, in
which the correlation kernels are expressed by the biorthogonal theta
functions. We demonstrate that they are elliptic extensions of the classical
determinantal point processes whose correlation kernels are expressed by
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