We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.
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.
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…
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.
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 $Π$.
For a broad class of point processes, including determinantal point processes, we construct associated marked and conditional ensembles, which allow to study a random configuration in the point process, based on information about a randomly incomplete part of the configuration. We show that our construction yields a well behaving transformation of sufficiently regular point processes. In the case of determinantal point processes, we explain that special cases of the conditional ensembles already appear implicitly in the literature, namely in the study of unitary invariant random matrix ensembl…
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.
For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity rho dnu, where nu is the corresponding invariant measure. We show that as rho converges to infinity, after centering, these processes converge to invariant H1 noise. More precisely, for all functions f in the interesection of H1(nu) and L1(nu) the distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with mean 0 and variance given by ||f||_H1^2 / (4 pi).
We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application, we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise.
We consider a family {P} of determinantal point processes arising in representation theory and random matrix theory. The processes live on the one-dimensional lattice and their correlation kernels correspond to projection operators in the l^2 Hilbert space on the lattice. Moreover, these projections are spectral projections associated to certain selfadjoint second order difference operators on the lattice. The aim of the note is to demonstrate that the difference operators in question can be efficiently employed in the study of limit transitions inside the family {P}.
Dynamical relationship between CAR algebras and determinantal point processes: point processes at finite temperature and stochastically positive KMS systems
Let "mu" be a point process on a countable discrete space "X". Under assumption that "mu" is quasi-invariant with respect to any finitary permutation of "X", we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which "mu" is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point processes "mu" possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection operator in the Hilbert space of square-summable functions on "X".
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 a work of Ghosh and Peres. 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 gene…
In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective char…
We consider smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. We prove asymptotic normality in the finite variance case, where Soshnikov's theorem is not applicable. The setting is similar to that of Rider and Virág [Electron. J. Probab., 12, no. 45, 1238--1257, (2007)] for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke Math J., 159, 31--81, (2011)] for eigenvalues of normal random…
We consider a discrete polynuclear growth (PNG) process and prove a functioal limit theorem for its convergrence to the Airy process. This generalizes previous results by Pr"ahofer and Spohn. The result enables us to express the GOE largest eigenvalue (Tracy-Widom) distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem.
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…
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 …
For a class of one-dimensional determinantal point processes including those induced by orthogonal projections with integrable kernels satisfying a growth condition, it is proved that their conditional measures, with respect to the configuration in the complement of a compact interval, are orthogonal polynomial ensembles with explicitly found weights. Examples include the sine-process and the process with the Bessel kernel. The argument uses the quasi-invariance, established in [1], of our point processes under the group of piecewise isometries of the real line.
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.
Consider Dyson's Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N tends to infinity. This limting determinantal proceess has the interesting feature that it shows number variance saturation. The variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity. Number variance saturation is also seen for example in the zeros of the Riemann zeta function (Odlyzko, Berry). The proce…
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter $\eps> 0$. The domain has two cusps, one pointing up and one pointing down. In the limit $\varepsilon\downarrow 0$ the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime $\varepsilon \downarrow 0$. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.
We show the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of $k$ independent $n\times n$ matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.
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 study determinantal point processes on $\mathbb{C}$ induced by the reproducing kernels of generalized Fock spaces as well as those on the unit disc $\mathbb{D}$ induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures $\textit{of the same order}$ are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures $\textit{of different orders}$ are singula…
The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on $\mathbb{S}^1$. It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to $\infty$. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Be…
By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unifi…
We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves according to random matrix theory, while, at large scales, it needs to be renormalized because the variance o…
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.
A point process is said to be rigid if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the complement of our bounded domain. The main result of this paper is that determinantal point processes with the Airy, the Bessel and the Gamma kernels are rigid. The proof follows the scheme of Ghosh [6], Ghosh and Peres [7]: the main step is the construction of a sequence of additive statistics with variance going to zero.
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…
The main result of this paper is that determinantal point processes on the real line corresponding to projection operators with integrable kernels are quasi-invariant, in the continuous case, under the group of diffeomorphisms with compact support (Theorem 1.4); in the discrete case, under the group of all finite permutations of the phase space (Theorem 1.6). The Radon-Nikodym derivative is computed explicitly and is given by a regularized multiplicative functional. Theorem 1.4 applies, in particular, to the sine-process, as well as to determinantal point processes with the Bessel and the Airy…
We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $\widetilde{O}(k^ω)$ time after an initial $\widetilde{O}(nk^{ω-1})$ time prep…
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 $λ\to \infty$.
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 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…
The $α$-determinant is a one-parameter generalisation of the standard determinant, with $α=-1$ corresponding to the determinant, and $α=1$ corresponding to the permanent. In this paper a simple limit procedure to construct $α$-determinantal point processes out of fermionic processes is examined. The procedure is illustrated for a model of $N$ free fermions in a harmonic potential. When the system is in the ground state, the rescaled correlation functions converge for large $N$ to determinants (of the sine kernel in the bulk and the Airy kernel at the edges). We analyse the point processes asso…
We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble --- a member of the new family. This allows us to obtain the…
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 study a class of stationary processes indexed by $\Z^d$ that are defined via minors of $d$-dimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong $K$ property, a particular strengthening of the usual $K$ (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to i.i.d. processes in the sense of ergodic theory). We obtain estimates of their entropi…
We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane C^+ associated with the ax+b (affine) group, depending on an admissible Hardy function ψ. We obtain the asymptotic behavior of the variance, the exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the variance on a compact set Ω contained in C^+. As a special case one recovers the DPP related to the weighted Bergman kernel. When ψ is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau lev…
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, ω)$ over a domain $U \subset \mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^\infty(U)$-module structure of $A^2(U, ω)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
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 study Palm measures of determinantal point processes with $J$-Hermitian correlation kernels. A point process $\mathbb{P}$ on the punctured real line $\mathbb{R}^* = \mathbb{R}_{+} \sqcup \mathbb{R}_{-}$ is said to be $\textit{balanced rigid}$ if for any precompact subset $B\subset \mathbb{R}^*$, the $\textit{difference}$ between the numbers of particles of a configuration inside $B\cap \mathbb{R}_{+}$ and $B\cap \mathbb{R}_{-}$ is almost surely determined by the configuration outside $B$. The point process $\mathbb{P}$ is said to have the $\textit{balanced Palm equivalence property}$ if any…
When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $ξ=\sum_{j \in Λ} δ_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $Φ$ having zeros in $\supp ξ$. Using such entire functions $Φ$, we define new topologies called the $Φ$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, …
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families ${A}_{N-1}$, ${B}_N$, ${C}_N$ and ${D}_N$, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dy…
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.
The (BC type) z-measures are a family of four parameter $z, z', a, b$ probability measures on the path space of the nonnegative Gelfand-Tsetlin graph with Jacobi-edge multiplicities. We can interpret the $z$-measures as random point processes $\mathcal{P}_{z, z', a, b}$ on the punctured positive real line $\mathfrak{X} = \mathbb{R}_+\setminus\{1\}$. Our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions. For very special values of the parameters $z, z'$, the processes $\mathcal{P}_{z, z'…
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.
Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functions
We show that the symplectic and orthogonal character analogues of Okounkov's Schur measure (on integer partitions) are determinantal, with explicit correlation kernels. We apply this to prove certain Borodin-Okounkov-Gessel-type results concerning Toeplitz+Hankel and Fredholm determinants; a Szegő-type limit theorem; an edge Baik-Deift-Johansson-type asymptotical result for certain symplectic and orthogonal analogues of the poissonized Plancherel measure; and a similar result for actual poissonized Plancherel measures supported on "almost symmetric" partitions.
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.
