Kurt Johansson
We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory.
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We give a complete description of bounded radial Toeplitz operators on the Hilbert space associated with a Cayley tree. As an application, we give a complete classification of a rich family of determinantal point processes on Cayley trees whose correlation kernels and thus distributions are invariant under the action of the graph automorphisms of the Cayley trees.
We introduce tree representations for $ α$-determinantal point processes. The $ α$-determinantal point processes is introduced as a one parameter extension of the determinantal point process. In the previous paper with H.Osada, the tree representation was introduced for determinantal point processes. In this paper, we prove that the tree representation can be applied to $ α$-determinantal point processes.
We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.
We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We give also some elementary, but nonetheless appealing relationship, between a discrete determinantal process and the well-known CS decomposition.
We give necessary and sufficient conditions for existence and infinite divisibility of $α$-determinantal processes. For that purpose we use results on negative binomial and ordinary binomial multivariate distributions.
We study some random interlaced configurations considering the eigenvalues of the main minors of Hermitian random matrices of the classical complex Lie algebras. We claim that these random configurations are determinantal and give their correlation kernels.
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
We describe the fundamental constructions and properties of determinantal probability measures and point processes, giving streamlined proofs. We illustrate these with some important examples. We pose several general questions and conjectures.
We study translation-invariant determinantal random point fields on the real line. We prove, under quite general conditions, that the smallest nearest spacings between the particles in a large interval have Poisson statistics as the length of the interval goes to infinity.
We consider a collection of Euclidean random balls in ${\Bbb R}^d$ generated by a determinantal point process inducing interaction into the balls. We study this model at a macros\-copic level obtained by a zooming-out and three different regimes --Gaussian, Poissonian and stable-- are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
We consider a collection of weighted Euclidian random balls in R^d distributed according a determinantal point process. We perform a zoom-out procedure by shrinking the radii while increasing the number of balls. We observe that the repulsion between the balls is erased and three different regimes are obtained, the same as in the weighted Poissonian case.
In this paper, we will derive the first and 2nd order Wiener chaos decomposition for the multivariate linear statistics of the determinantal point processes associated with the spectral projection kernels on the unit spheres $S^d$. We will first get a graphical representation for the cumulants of multivariate linear statistics for any determinantal point process. The main results then follow from the very precise estimates and identities regarding the spectral projection kernels and the symmetry of the spheres.
We derive an integration by parts formula for functionals of determinantal processes on compact sets, completing the arguments of [4]. This is used to show the existence of a configuration-valued diffusion process which is non-colliding and admits the distribution of the determinantal process as reversible law. In particular, this approach allows us to build a concrete example of the associated diffusion process, providing an illustration of the results of [4] and [30].
The CS decomposition and conditional negative correlation inequalities for determinantal processes
本文研究行列式过程在排除集合包含条件下的负相关性，借助子空间的主向量、主角及 CS 分解揭示其几何结构，并证明新的条件负相关不等式。
We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an…
Fluctuations of two-dimensional determinantal processes associated with Berezin--Toeplitz operators
本文研究由Berezin--Toeplitz算子自由费米子基态导出的新行列式点过程，推广Ginibre系综，证明Szegő型展开并得到中心极限定理，揭示边界波动依赖哈密顿动力学且非共形不变。
We extend Lyons's tree entropy theorem to general determinantal measures. As a byproduct we show that the sofic entropy of an invariant determinantal measure does not depend on the chosen sofic approximation.
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 and Shirai and Takahashi. As its continuum version, we prove an isomorphism between the translation-invariant determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels and homogeneous Poisson point processes. For this purpose, we also prove the Bernoulli property for the tree representations of the dete…
Functional description of a class of quasi-invariant determinantal processes
该研究利用整函数的de Branges空间，建立了对应投影核的一类拟不变行列式过程的泛函表征，推进了相关随机过程的泛函描述研究。
We give a simple algebraic derivation of a useful determinantal identity for multilevel systems such as random matrix chains and finite determinantal point processes, with applications to the calculation of point correlators, gap probabililties and Janossy densities.
We compute the expected Riesz energy of random points on flat tori drawn from certain translation invariant determinantal processes and determine the process in the family providing the optimal asymptotic expected Riesz energy.
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae, and that any one-dependent point process on the integers is determinantal. The examples give a gentle introduction to the emerging fields of one-dependent an…
We present a list of algebraic, combinatorial, and analytic mechanisms that give rise to determinantal point processes.
Determinantal process is a dynamical extension of a determinantal point process such that any spatio-temporal correlation function is given by a determinant specified by a single continuous function called the correlation kernel. Noncolliding diffusion processes are important examples of determinantal processes. In the present lecture, we introduce determinantal martingales and show that if the interacting particle system (IPS) has determinantal-martingale representation, then it becomes a determinantal process. From this point of view, the reason why noncolliding diffusion processes and nonco…
Let $F$ be a non-discrete non-Archimedean local field. For any subset $S\subset F$ with finite Haar measure, there is a stationary determinantal point process on $F$ with correlation kernel $\widehat{\mathbb{1}}_S(x-y)$, where $\widehat{\mathbb{1}}_S$ is the Fourier transform of the indicator function $\mathbb{1}_S$. In this note, we give a geometrical condition on the subset $S$, such that the associated determinantal point process is rigid in the sense of Ghosh and Peres. Our geometrical condition is very different from the Euclidean case.
Product approximations for uniform spanning trees and determinantal processes
该研究针对有限连通简单图的均匀生成树与随机1-出子图建立耦合，推广至行列式过程与1-出过程的耦合，推导了相关同调挠率的指数增长率。
We construct a canonical embedding of the space $L^2$ over a determinantal point process to the fermionic Fock space. Equivalently, we show that a determinantal process is the spectral measure for some explicit commutative group of Gaussian operators in the fermionic Fock space.
We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.
Noncolliding Brownian motion (Dyson's Brownian motion model with parameter $β=2$) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy ker…
On distances among Slater Determinant States and Determinantal Point Processes
本文研究Slater行列式态与行列式点过程间的距离，通过迹/总变分和Wasserstein距离建立定量界限，揭示两者结构的联系。
We prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized Gaussian random process.
Determinantal point processes on a measure space X whose kernels represent trace class Hermitian operators on L^2(X) are associated to "quasifree" density operators on the Fock space over L^2(X).
The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point processes satisfy the so-called condition $(Σ_λ)$ which is a general form of Gibbsianness. Under a continuity assumption, the Gibbsian conditional probabilities can be identified explicitly.
The main result of this paper states that if $(N, Π)$ is a pair of independent point processes on a common ground space with $N$ Poisson and $Π$ determinantal induced by a locally trace class (not necessarily self-adjoint) correlation kernel, then their independent superposition $N + Π$ determines uniquely the distributions of $N$ and $Π$.
We propose a new class of algorithms for randomly scheduling network transmissions. The idea is to use (discrete) determinantal point processes (subsets) to randomly assign medium access to various {\em repulsive} subsets of potential transmitters. This approach can be seen as a natural extension of (spatial) Aloha, which schedules transmissions independently. Under a general path loss model and Rayleigh fading, we show that, similarly to Aloha, they are also subject to elegant analysis of the coverage probabilities and transmission attempts (also known as local delay). This is mainly due to t…
In this note we consider sampling from (non-homogeneous) strongly Rayleigh probability measures. As an important corollary, we obtain a fast mixing Markov Chain sampler for Determinantal Point Processes.
Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional independence. We describe some conditional independencies through the conditions on the kernel of a determinantal point process, and show many can be obtained using the graph induced by a kernel of the $L$-ensemble.
Strongly Rayleigh distributions are natural generalizations of product and determinantal probability distributions and satisfy strongest form of negative dependence properties. We show that the "natural" Monte Carlo Markov Chain (MCMC) is rapidly mixing in the support of a {\em homogeneous} strongly Rayleigh distribution. As a byproduct, our proof implies Markov chains can be used to efficiently generate approximate samples of a $k$-determinantal point process. This answers an open question raised by Deshpande and Rademacher.
The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the determinantal random point field with Hermitian kernel and a criterion for the weak convergence of its distribution. In the second section we proceed with the examples of the determinantal random point fields from Quantum Mechanics, Statistical Mechanics, Random Matrix Theory, Probability Theory, Representation Theory and Ergodic Theory. In connection with the Th…
Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in the sense that any spatio-temporal correlation function can be expressed by a determinant. The purpose of the present paper is to clarify the connection between these two aspects. We introduce a notion of determinantal martingale and prove that, if the system has determinantal-martingale representation, then it is determinantal. In order to demonstrate the …
Likelihood Geometry of Determinantal Point Processes
本文利用代数统计方法研究行列式点过程的似然几何，通过计数并计算小模型的对数似然临界点，反驳了Brunel等人提出的一个猜想。
In this note, we show that determinantal point processes on the real line corresponding to de Branges spaces of entire functions are rigid in the sense of Ghosh-Peres and, under certain additional assumptions, quasi-invariant under the group of diffeomorphisms of the line with compact support.
We consider the Laplace-Beltrami operator $Δ_g$ on a smooth, compact Riemannian manifold $(M,g)$ and the determinantal point process $\mathcal{X}_λ$ on $M$ associated with the spectral projection of $-Δ_g$ onto the subspace corresponding to the eigenvalues up to $λ^2$. We show that the pull-back of $\mathcal{X}_λ$ by the exponential map $\exp_p : T_p^*M \to M$ under a suitable scaling converges weakly to the universal determinantal point process on $T_p^* M$ as $λ\to \infty$.
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