scholarly journals Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles

2020 ◽  
Vol 21 (12) ◽  
pp. 3973-4002
Author(s):  
Gernot Akemann ◽  
Eugene Strahov ◽  
Tim R. Würfel
Keyword(s):  
External Field ◽  
Random Matrices ◽  
Point Processes ◽  
Probability Measures ◽  
Characteristic Polynomials ◽  
Quantum Field Theories ◽  
Determinantal Point Processes ◽  
Schur Polynomials ◽  
Expectation Values ◽  
Ginibre Ensemble

Abstract Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

scholarly journals Determinantal point processes in the plane from products of random matrices

2016 ◽  
Vol 52 (1) ◽  
pp. 16-46 ◽  
Author(s):  
Kartick Adhikari ◽  
Nanda Kishore Reddy ◽  
Tulasi Ram Reddy ◽  
Koushik Saha
Keyword(s):  
Random Matrices ◽  
Point Processes ◽  
Determinantal Point Processes ◽  
Products Of Random Matrices

scholarly journals Characteristic Polynomials of Complex Random Matrices and Painlevé Transcendents

2020 ◽  
Author(s):  
Alfredo Deaño ◽  
Nick Simm
Keyword(s):  
Correlation Function ◽  
Random Matrices ◽  
Characteristic Polynomials ◽  
Painlevé Transcendents ◽  
Ginibre Ensemble ◽  
Painleve Transcendents ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Non Gaussian ◽  
Unitary Ensemble

Abstract We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite $N$ and asymptotically as $N \to \infty $. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlevé IV at the boundary as $N \to \infty $. Our approach, together with the results in [ 49], suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two “planar Fisher–Hartwig singularities” where Painlevé V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with $d$-fold rotational symmetries known as the lemniscate ensemble, recently studied in [ 15, 18]. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlevé VI arises at finite $N$. Scaling near the boundary leads to Painlevé V, in contrast to the Ginibre ensemble.

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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