Determinantal Processes Research Articles

Functional Description of a Class of Quasi-Invariant Determinantal Processes

Functional Description of a Class of Quasi-Invariant Determinantal Processes

Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 4$$\\end{document}, which appear new for n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 5$$\\end{document}. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

Optimal semiclassical regularity of projection operators and strong Weyl law

Projection operators arise naturally as one-particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in Chong et al. [J. Eur. Math. Soc. (unpublished) (2024)] about the possibility of having projection operators as initial data. It also gives a strong convergence result in Sobolev spaces for the Weyl law in phase space.

Boundary asymptotics of non-intersecting Brownian motions: Pearcey, Airy and a transition

We study n non-intersecting Brownian motions, corresponding to the eigenvalues of an n × n Hermitian Brownian motion. At the boundary of their limit shape we find that only three universal processes can arise: the Pearcey process close to merging points, the Airy line ensemble at edges and a novel determinantal process describing the transition from the Pearcey process to the Airy line ensemble. The three cases are distinguished by a remarkably simple integral condition. Our results hold under very mild assumptions, in particular we do not require any kind of convergence of the initial configuration as n→∞ . Applications to largest eigenvalues of macro- and mesoscopic bulks and to random initial configurations are given.

On tail triviality of negatively dependent stochastic processes

We prove that every negatively associated sequence of Bernoulli random variables with “summable covariances” has a trivial tail σ-field. A corollary of this result is the tail triviality of strongly Rayleigh processes. This is a generalization of a result due to Lyons, which establishes tail triviality for discrete determinantal processes. We also study the tail behavior of negatively associated Gaussian and Gaussian threshold processes. We show that these processes are tail trivial though, in general, they do not satisfy the summable covariances property. Furthermore, we construct negatively associated Gaussian threshold vectors that are not strongly Rayleigh. This identifies a natural family of negatively associated measures that is not a subset of the class of strongly Rayleigh measures.

Matchings on trees and the adjacency matrix: A determinantal viewpoint

AbstractLet be a finite tree. For any matching of , let be the set of vertices uncovered by . Let be a uniform random maximum size matching of . In this paper, we analyze the structure of . We first show that is a determinantal process. We also show that for most vertices of , the process in a small neighborhood of that vertex can be well approximated based on a somewhat larger neighborhood of the same vertex. Then we show that the normalized Shannon entropy of can be also well approximated using the local structure of . In other words, in the realm of trees, the normalized Shannon entropy of —that is, the normalized logarithm of the number of maximum size matchings of —is a Benjamini‐Schramm continuous parameter.

Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

A note related to the CS decomposition and the BK inequality for discrete determinantal processes

AbstractWe prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

Ergodic quasi-exchangeable stationary processes are isomorphic to Bernoulli processes

A discrete time process, with law $$\mu $$ , is quasi-exchangeable if for any finite permutation $$\sigma $$ of time indices, the law $$\mu ^\sigma $$ of the resulting process is equivalent to $$\mu $$ . For a quasi-exchangeable stationary process, our main results are (1) if the process is ergodic then it is isomorphic to a Bernoulli process and (2) if the family of all Radon–Nikodym derivatives $$\{{d\mu ^\sigma \over d\mu }\}$$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti’s Theorem. We give application of (1) to some determinantal processes.

Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques

Spatial statistics is a powerful tool for analyzing data that are illustrated as points or positions in a regular or non-regular state space. Techniques that are proposed to investigate the spatial association between neighboring positions are based on the point process analysis. One of the main goals is to simulate real data positions (such as antenna base stations) using the type of point process that most closely matches the data. Spatial patterns could be detailed describing the observed positions and appropriate models were proposed to simulate these patterns. A common model to simulate spatial patterns is the Poisson point process. In this work analyses of the Poisson point process—as well as modified types such as inhibition point process and determinantal Poisson point process—are presented with simulated data close to the true data (i.e., antenna base station positions). Investigation of the spatial variation of the data led us to the spatial association between positions by applying Ripley’s K-functions and L-Function.

Stationary determinantal processes: [formula omitted]-mixing property and correlation dimensions

We study the stationary determinantal processes on the integer lattice, induced by strictly positive and strictly contractive convolution kernels. A necessary and sufficient condition of ψ-mixing property for such a process is given. Some new results on the correlation dimension of the stationary determinantal measure on the symbolic space are obtained. As an application, the precise decreasing rate of the shortest distances between two orbits in the corresponding dynamical system is obtained.

Paving property for real stable polynomials and strongly Rayleigh processes

One of the equivalent formulations of the Kadison–Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the “paving conjecture”. In this paper, we first extend this result to real stable polynomials. We prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the “restriction” of the polynomial to each subset has small roots. Then, we derive a probabilistic interpretation of this result. We show that there exists a partition of the underlying space of every strongly Rayleigh process into a small number of sets such that the restriction of the point process to each set has “almost independent” points. This result implies that the dependence structure of strongly Rayleigh processes is constrained—a phenomenon that is to be expected in negatively dependent measures. To prove this result, we introduce and study the notion of kernel polynomial for strongly Rayleigh processes. This notion is a natural generalization of the kernel of determinantal processes. We also derive an entropy bound for strongly Rayleigh processes in terms of the roots of the kernel polynomial which is interesting on its own.

Scaling Limits of Planar Symplectic Ensembles

We consider various asymptotic scaling limits $N\to\infty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials $Q(\zeta)=|\zeta|^{2\lambda}-(2c/N)\log|\zeta|$, with $\lambda>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/\lambda$ at the origin. For integer $m=1/\lambda$ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain.

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Given a sequence \((\xi _n)\) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper “Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process” Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series \(f(z) = \sum _{n=1}^\infty \xi _n z^{n-1}\) in the complex unit disc \({\mathbb {D}}\) constitute a determinantal point process. The study of the zeros of the general random series f(z), where the restriction of independence is relaxed upon the random variables \((\xi _n)\) is an important open problem. This paper proves that if \((\xi _n)\) is an infinite sequence of complex Gaussian random variables, such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f(z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

Gap probability for products of random matrices in the critical regime

The singular values of a product of M independent Ginibre matrices of size N×N form a determinantal point process. Near the soft edge, as both M and N go to infinity in such a way that M/N→α, α>0, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval (a,+∞). We derive a Tracy–Widom-like formula in terms of the unique solution of a certain matrix Riemann–Hilbert problem of size 2 × 2. The right-tail asymptotics for this solution is obtained by the Deift–Zhou non-linear steepest descent analysis.

Rigidity Hierarchy in Random Point Fields: Random Polynomials and Determinantal Processes

In certain point processes, the configuration of points outside a bounded domain determines, with probability 1, certain statistical features of the points within the domain. This notion, called rigidity, was introduced in Ghosh and Peres (Duke Math J 166(10):1789–1858, 2017). In this paper, rigidity and the related notion of tolerance are examined systematically and point processes with rigidity of various degrees are introduced. Natural classes of point processes such as determinantal point processes, zero sets of Gaussian entire functions and perturbed lattices are examined from the point of view of rigidity, and general conditions are provided for them to exhibit specified nature of spatially rigid behaviour. In particular, we examine the rigidity of determinantal point processes in terms of their kernel, and demonstrate that a necessary condition for determinantal processes to exhibit rigidity is that their kernel must be a projection. We introduce a one parameter family of point processes which exhibit arbitrarily high levels of rigidity (depending on the choice of parameter value), answering a natural question on point processes with higher levels of rigidity (beyond the known examples of rigidity of local mass and center of mass). Our one parameter family is also related to a natural extension of the standard planar Gaussian analytic function process and their zero sets.

Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential

The random variable 1 + z 1 + z 1 z 2 + … appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to N × N matrices either real symmetric β = 1 or complex Hermitian β = 2. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to N fermions in a Morse potential, which are non-interacting for β = 2. At finite N the distribution of eigenvalues exhibits heavy tails, generalizing Kesten’s results in the scalar case. The density of fermions in this potential is studied for large N, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large N limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valkó, Grabsch and Texier, as well as Ossipov, is discussed.

Precise Deviations for Disk Counting Statistics of Invariant Determinantal Processes

Abstract We consider 2-dimensional determinantal processes that are rotationinvariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry–Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal processes, including the polyanalytic Ginibre ensembles, zeros of the hyperbolic Gaussian analytic function, and other hyperbolic models. As a corollary, we compute the precise asymptotics for the entanglement entropy of (integer) Laughlin states for all Landau levels.

Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ with N Labeled Objects per Site, Part I: Basic Properties and Full Domination

Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ with N Labeled Objects per Site, Part I: Basic Properties and Full Domination

Determinantal Point Processes for Image Processing

Determinantal point processes (DPPs) are probabilistic models of configurations that favor diversity or repulsion. They have recently gained influence in the machine learning community, mainly because of their ability to elegantly and efficiently subsample large sets of data. In this paper, we consider DPPs from an image processing perspective, meaning that the data we want to subsample are pixels or patches of a given image. To this end, our framework is discrete and finite. First, we adapt their basic definition and properties to DPPs defined on the pixels of an image, that we call determinantal pixel processes (DPixPs). We are mainly interested in the repulsion properties of such a process and we apply DPixPs to texture synthesis using shot noise models. Finally, we study DPPs on the set of patches of an image. Because of their repulsive property, DPPs provide a strong tool to subsample discrete distributions such as that of image patches.