scholarly journals Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes

2019 ◽  
Vol 60 (1) ◽  
pp. 013301 ◽  
Author(s):  
Makoto Katori
Keyword(s):  
Point Processes ◽  
Theta Functions ◽  
Root Systems ◽  
Determinantal Point Processes ◽  
Affine Root Systems

scholarly journals The Elliptic Tail Kernel

2020 ◽  
Author(s):  
Cesar Cuenca ◽  
Vadim Gorin ◽  
Grigori Olshanski
Keyword(s):  
Harmonic Analysis ◽  
Point Processes ◽  
Theta Functions ◽  
Young Diagrams ◽  
Determinantal Point Processes ◽  
Translation Invariant ◽  
New Family ◽  
Jacobi Theta Functions ◽  
Correlation Kernels

Abstract We introduce and study a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice. We prove that these processes are limits of the $q$–$zw$ measures, which arise in the $q$-deformation of harmonic analysis on $U(\infty )$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $q$–$zw$ measures are diffuse. Our results also hint at a link between the two-sided $q$-lattice and rows/columns of Young diagrams.

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

2008 ◽  
Vol 04 (03) ◽  
pp. 461-474 ◽  
Author(s):  
PEE CHOON TOH
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Root Systems ◽  
Jacobi Theta Functions ◽  
Multiple Variables ◽  
Affine Root Systems ◽  
Theta Function Identities

We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

2021 ◽  
Vol 58 (2) ◽  
pp. 469-483
Author(s):  
Jesper Møller ◽  
Eliza O’Reilly
Keyword(s):  
Finite Number ◽  
Point Process ◽  
Point Processes ◽  
Parametric Models ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
The Difference ◽  
Weaker Conditions ◽  
Palm Distributions

AbstractFor a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

scholarly journals High-performance sampling of generic determinantal point processes

2020 ◽  
Vol 378 (2166) ◽  
pp. 20190059 ◽  
Author(s):  
Jack Poulson
Keyword(s):  
Point Process ◽  
Spectral Decomposition ◽  
High Performance ◽  
Point Processes ◽  
Cholesky Factorization ◽  
Determinantal Point Processes ◽  
Determinantal Point Process ◽  
Decomposition Approach ◽  
Map Inference ◽  
Sampling Schemes

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7 , 83–122) as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system (Kulesza & Taskar 2012 Found. Trends Mach. Learn. 5 , 123–286). The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive set-up cost. Researchers Launay et al. 2018 ( http://arxiv.org/abs/1802.08429 ); Chen & Zhang 2018 NeurIPS ( https://papers.nips.cc/paper/7805-fast-greedy-map-inference-for-determinantal-point-process-to-improve-recommendation-diversity.pdf ) have begun to connect DPP sampling to LDL H factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. This article proves that trivial modifications of LU and LDL H factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Furthermore, it is experimentally shown that even dynamically scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. The software developed as part of this research, Catamari ( hodgestar.com/catamari ) is released under the Mozilla Public License v.2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

scholarly journals On a few statistical applications of determinantal point processes

2017 ◽  
Vol 60 ◽  
pp. 180-202 ◽  
Author(s):  
Rémi Bardenet ◽  
Frédéric Lavancier ◽  
Xavier Mary ◽  
Aurélien Vasseur
Keyword(s):  
Point Processes ◽  
Determinantal Point Processes
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