Determinantal point processes with J-Hermitian correlation kernels

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let $\Gamma_X$ denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on $\Gamma_X$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n)}(x_1,\ldots,x_n)=\det[K(x_i,x_j)]_{i,j=1,\ldots,n}$. The function K(x,y) is called the correlation kernel of the determinantal point process $\mu$. Assume that the space X is split into two parts: $X=X_1\sqcup X_2$. A kernel K(x,y) is called J-Hermitian if it is Hermitian on $X_1\times X_1$ and $X_2\times X_2$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_1$ and $y\in X_2$. We derive a necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel K(x,y).Comment: Published in at http://dx.doi.org/10.1214/12-AOP795 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR

Let $X$ be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators $a(\Delta)$ indexed by all measurable, relatively compact sets $\Delta$ in $X$ (a quantum stochastic process over $X$). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over $X$ having the same correlation measure. Furthermore, the operators $a(\Delta)$ can be realized as multiplication operators in the $L^2$-space with respect to this point process. In the proof, we utilize the notion of $\star$-positive definiteness, proposed in [Y. G. Kondratiev and T.\ Kuna, {\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\bf 5} (2002), 201--233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space $X$, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way

Equilibrium Kawasaki dynamics and determinantal point processes

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".Comment: 13 pp; to appear in J. Math. Sci. (New York

Particle-Hole Transformation in the Continuum and Determinantal Point Processes

Let X be an underlying space with a reference measure σ. Let K be anintegral operator in L2(X,σ) with integral kernel K(x, y). A point process μ on X iscalled determinantal with the correlation operator K if the correlation functions of μ aregiven by k(n)(x1,..., xn) = det[K(xi, x j)]i,j=1,...,n. It is known that each determinantalpoint process with a self-adjoint correlation operator K is the joint spectral measure of theparticle density ρ(x) = A+(x)A−(x) (x ∈ X), where the operator-valued distributionsA+(x), A−(x) come from a gauge-invariant quasi-free representation of the canonicalanticommutation relations (CAR). If the space X is discrete and divided into two disjointparts, X1 and X2, by exchanging particles and holes on the X2 part of the space, oneobtains from a determinantal point process with a self-adjoint correlation operator Kthe determinantal point process with the J -self-adjoint correlation operator K = K P1 +(1 − K)P2. Here Pi is the orthogonal projection of L2(X,σ) onto L2(Xi,σ). In thecase where the space X is continuous, the exchange of particles and holes makes nosense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-freerepresentation of the CAR. This transformation acts identically on the X1 part of thespace and exchanges the creation operators A+(x) and the annihilation operators A−(x)for x ∈ X2. This leads to a quasi-free representation of the CAR, which is not anymoregauge-invariant. We prove that the joint spectral measure of the corresponding particledensity is the determinantal point process with the correlation operator K