Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Macdonald denominators for affine root systems, orthogonal theta
  functions, and elliptic determinantal point processes   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
trigonometric and rational functions. In the scaling limits associated with $N
\to \infty$, we obtain four types of elliptic determinantal point processes
with an infinite number of points and parameter $t \in (0, t_{\ast})$. We give
new expressions for the Macdonald denominators using the
Karlin--McGregor--Lindstr\"om--Gessel--Viennot determinants for noncolliding
Brownian paths, and show the realization of the associated elliptic
determinantal point processes as noncolliding Brownian brides with a time
duration $t_{\ast}$, which are specified by the pinned configurations at time
$t=0$ and $t=t_{\ast}$.

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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con