Determinantal Processes Research Articles

Construction of point processes for classical and quantum gases

We propose a construction of point processes via the method of cluster expansion. The important role of the class of infinitely divisible point processes is noted. Examples are permanental and determinantal processes as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.

Hole Probabilities and Overcrowding Estimates for Products of Complex Gaussian Matrices

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda J. Phys A: Math. Theor. 45 (2012) 465201), which can be understood as a generalization of the finite Ginibre ensemble. As N\rightarrow\infty, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as {R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2 is distributed as the product of n independent Gamma variables Gamma(k,1). This enables us to find the asymptotics for the hole probabilities, i.e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r\rightarrow\infty. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m\rightarrow\infty.

Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles

The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group $S_\infty$. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz matrices of infinite size. The purpose of the paper is to construct a family $\{X^{(z,z')}\}$ of continuous time Markov processes on the Thoma cone, depending on two continuous parameters $z$ and $z'$. Our construction largely exploits specific properties of the Thoma cone related to its representation-theoretic origin, although we do not use representations directly. On the other hand, we were inspired by analogies with random matrix theory coming from models of Markov dynamics related to orthogonal polynomial ensembles. We show that processes $X^{(z,z')}$ possess a number of nice properties, namely: (1) every $X^{(z,z')}$ is a Feller process; (2) the infinitesimal generator of $X^{(z,z')}$, its spectrum, and the eigenfunctions admit an explicit description; (3) in the equilibrium regime, the finite-dimensional distributions of $X^{(z,z')}$ can be interpreted as (the laws of) infinite-particle systems with determinantal correlations; (4) the corresponding time-dependent correlation kernel admits an explicit expression, and its structure is similar to that of time-dependent correlation kernels appearing in random matrix theory.

System of Complex Brownian Motions Associated with the O’Connell Process

The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length $a$. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is $N$, the rank of the matrix of the Fredholm determinant is $N$. Then we give a representation for the quantity by using an $N$-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit $a \to 0$ it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.

The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap \(\lambda _{i+1}(M_n)-\lambda _i(M_n)\) of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin–Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin–Mehta law required either an averaging in the eigenvalue index parameter \(i\), or fixing the energy level \(u\) instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function \(N_{(-\infty ,x)}(\tilde{M}_n)\) (where \(\tilde{M}_n\) is a suitably rescaled version of \(M_n\)) with the event that there is no spectrum in an interval \([x,x+s]\), in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.

A representation of the moment measures of the general ideal Boe gas

AbstractWe reconsider the fundamental work of Fichtner 2 and exhibit the permanental structure of the ideal Bose gas again, using a new approach which combines a characterization of infinitely divisible random measures (due to Kerstan, Kummer and Matthes 4, 6 and Mecke 9, 10) with a decomposition of the moment measures into its factorial measures due to Krickeberg 5. To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain “loop” integrals. This representation can be considered as a point process analogue of the old idea of Symanzik 15 that local times and self‐crossings of the Brownian motion can be used as a tool in quantum field theory.Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Lévy‐measure belonging to some large class of measures containing that of the classical ideal Bose gas considered by Fichtner.It is well‐known that the calculation of moments of higher order of point processes is notoriously complicated. See for instance Krickeberg’s calculations for the Poisson or the Cox process in 5.Relations to the work of Shirai, Takahashi 12 and Soshnikov 14 on permanental and determinantal processes are outlined.

Determinantal Process Starting from an Orthogonal Symmetry is a Pfaffian Process

When the number of particles $N$ is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index $\nu > -1$ (BESQ$^{(\nu)}$) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The $2 \times 2$ skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, $N \delta_0$, and by the equivalence between the noncolliding BESQ$^{(\nu)}$ and that of the noncolliding squared generalized meander starting from $N \delta_0$.

Spanning forests and the vector bundle Laplacian

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.

Limits of determinantal processes near a tacnode

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter e > 0. The domain has two cusps, one pointing up and one pointing down. In the limit e ↓ 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 e ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

Correlation Inequalities for Edge-Reinforced Random Walk

We prove correlation inequalities for linearly edge-reinforced random walk. These correlation inequalities concern the first entry tree, i.e. the tree of edges used to enter any vertex for the first time. They also involve the asymptotic fraction of time spent on particular edges. Basic ingredients are known FKG-type inequalities and known negative associations for determinantal processes.

Recursive equations for the predictive distributions of some determinantal processes

This paper provides recursive equations for the predictive distributions of one-dependent and two-dependent determinantal processes. Fixed order recursive equations can be applied both to efficiently simulate trajectories and to explore properties of the process.

Orbit measures, random matrix theory and interlaced determinantal processes

A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal.

Quasi-invariance and integration by parts for determinantal and permanental processes

Determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent. Their atoms exhibit mutual attraction of repulsion, thus these processes are very far from the uncorrelated situation encountered in Poisson models. We establish a quasi-invariance result: we show that if atom locations are perturbed along a vector field, the resulting process is still a determinantal (respectively permanental) process, the law of which is absolutely continuous with respect to the original distribution. Based on this formula, following Bismut approach of Malliavin calculus, we then give an integration by parts formula.

On adding a list of numbers (and other one-dependent determinantal processes)

Adding a column of numbers produces 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 and determinantal point processes.

非衝突過程・行列値過程・行列式過程

非衝突過程・行列値過程・行列式過程

Stochastic Analysis

Stochastic Analysis

From random matrices to random analytic functions

We consider two families of random matrix-valued analytic functions: (1) G1−zG2 and (2) G0+zG1+z2G2+⋯, where Gi are n×n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces (“Bargmann–Fock spaces”) of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virág (n=1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

Point processes in arbitrary dimension from fermionic gases, random matrix theory, andnumber theory

It is well known that one can map certain properties of random matrices, fermionic gases,and zeros of the Riemann zeta function to a unique point process on the real line . Here we analytically provide exact generalizations of such a point process ind-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain then-particle correlationfunctions for any n, which completely specify the point processes in . We also demonstrate that spin-polarized fermionic systems in have these same n-particle correlation functions in each dimension. The point processes for anyd are shown to be hyperuniform, i.e., infinite wavelength densityfluctuations vanish, and the structure factor (or power spectrum)S(k) has a non-analytic behavior at the origin given byS(k)∼|k| (). The latter result implies that the pair correlation functiong2(r) tends to unity for large pair distances with a decay rate that is controlled by the power law1/rd+1, which is a well-known property of bosonic ground states and more recently has beenshown to characterize maximally random jammed sphere packings. We graphically displayone-and two-dimensional realizations of the point processes in order to vividlyreveal their ‘repulsive’ nature. Indeed, we show that the point processes can becharacterized by an effective ‘hard core’ diameter that grows like the square root ofd. The nearest-neighbor distribution functions for these point processes arealso evaluated and rigorously bounded. Among other results, this analysisreveals that the probability of finding a large spherical cavity of radiusr indimension d behaves like a Poisson point process but in dimensiond+1, i.e., this probabilityis given by exp[−κ(d)rd+1] for large r andfinite d, whereκ(d) is a positived-dependent constant.We also show that as d increases, the point process behaves effectively like a spherepacking with a coverage fraction of space that is no denser than1/2d. This coverage fraction has a special significance in the study of sphere packings inhigh-dimensional Euclidean spaces.

Nonintersecting paths and the Hahn orthogonal polynomial ensemble

We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation-invariant extension of the discrete sine process, which can also be described by an ergodic Gibbs measure with appropriate parameters.

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.