84 research outputs found
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 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 . A point process is called
determinantal if its correlation functions have the form
. The function
K(x,y) is called the correlation kernel of the determinantal point process
. Assume that the space X is split into two parts: . A
kernel K(x,y) is called J-Hermitian if it is Hermitian on and
, and for and .
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 be a locally compact, second countable Hausdorff topological space.
We consider a family of commuting Hermitian operators indexed by
all measurable, relatively compact sets in (a quantum stochastic
process over ). 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 having the same correlation measure. Furthermore, the
operators can be realized as multiplication operators in the
-space with respect to this point process. In the proof, we utilize the
notion of -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 , 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 Ο. Let K be anintegral operator in L2(X,Ο) with integral kernel K(x, y). A point process ΞΌ on X iscalled determinantal with the correlation operator K if the correlation functions of ΞΌ 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 Ο(x) = A+(x)Aβ(x) (x β X), where the operator-valued distributionsA+(x), Aβ(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 β K)P2. Here Pi is the orthogonal projection of L2(X,Ο) onto L2(Xi,Ο). 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 Aβ(x)for x β 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
- β¦