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13
Gibbsianness of Fermion Random Point Fields
, 2005
"... We consider fermion (or determinantal) random point fields on Euclidean space Rd. Given a bounded, translation invariant, and positive definite integral operator J on L2 (Rd), we introduce a determinantal interaction for a system of particles moving on Rd as follows: the n points located at x1, · ..."
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We consider fermion (or determinantal) random point fields on Euclidean space Rd. Given a bounded, translation invariant, and positive definite integral operator J on L2 (Rd), we introduce a determinantal interaction for a system of particles moving on Rd as follows: the n points located at x1, · · · , xn ∈ Rd have the potential energy given by U (J) (x1, · · · , xn): = − log det(j(xi − xj))1≤i,j≤n, where j(x − y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is welldefined. When J is of finite range in addition, and for d ≥ 2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I + J) −1 is a Gibbs measure admitted to the specification.
Statistics of Extreme Spacings in Determinantal Random Point Processes
, 2006
"... Determinantal (a.k.a. fermion) random point processes were introduced in probability theory by Macchi about thirty years ago ([13], [14], [3]). In the ..."
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Determinantal (a.k.a. fermion) random point processes were introduced in probability theory by Macchi about thirty years ago ([13], [14], [3]). In the
A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application
, 2005
"... Given a positive definite, bounded linear operator A on the Hilbert space H0: = l 2 (E), we consider a reproducing kernel Hilbert space H+ with a reproducing kernel A(x, y). Here E is any countable set and A(x, y), x, y ∈ E, is the representation of A w.r.t. the usual basis of H0. Imposing further c ..."
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Given a positive definite, bounded linear operator A on the Hilbert space H0: = l 2 (E), we consider a reproducing kernel Hilbert space H+ with a reproducing kernel A(x, y). Here E is any countable set and A(x, y), x, y ∈ E, is the representation of A w.r.t. the usual basis of H0. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space H− with a kernel function B(x, y), which is the representation of the inverse of A in a sense, so that H − ⊃ H0 ⊃ H+ becomes a rigged Hilbert space. We investigate a relationship between the ratios of determinants of some partial matrices related to A and B and the suitable projections in H − and H+. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I + A) −1 on the set E. It turns out that the class of determinantal point processes that can be recognized as Gibbs measures for suitable interactions is much bigger than that obtained by Shirai and Takahashi.
A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes
, 2007
"... We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space X, for which certain fermion point processes are invariant. The Glauber dynamics is a birthanddeath process in X, while in the case of the Kawasaki dynamics interacting particles randomly ..."
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Cited by 1 (1 self)
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We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space X, for which certain fermion point processes are invariant. The Glauber dynamics is a birthanddeath process in X, while in the case of the Kawasaki dynamics interacting particles randomly hop over X. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.
Discussion of ‘Modern Statistics for Spatial Point Processes’
"... ABSTRACT. The paper ‘Modern statistics for spatial point processes ’ by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st ..."
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ABSTRACT. The paper ‘Modern statistics for spatial point processes ’ by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st
Properties and simulation of αpermanental random fields
"... An αpermanental random field is briefly speaking a model for a collection of random variables with positive associations, where α is a positive number and the probability generating function is given in terms of a covariance or more general function so that density and moment expressions are given ..."
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An αpermanental random field is briefly speaking a model for a collection of random variables with positive associations, where α is a positive number and the probability generating function is given in terms of a covariance or more general function so that density and moment expressions are given by certain αpermanents. Though such models possess many appealing probabilistic properties, many statisticians seem unaware of αpermanental random fields and their potential applications. The purpose of this paper is first to summarize useful probabilistic results using the simplest possible setting, and second to study stochastic constructions and simulation techniques, which should provide a useful basis for discussing the statistical aspects in future work. The paper also discusses some examples of αpermanental random fields.
EXTREME GAPS BETWEEN EIGENVALUES OF RANDOM MATRICES
, 2011
"... This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haardistributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the kth smallest gap, normalized by a factor n 4/3, has a limiting d ..."
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This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haardistributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the kth smallest gap, normalized by a factor n 4/3, has a limiting density proportional to x 3k−1 e −x3. Concerning the largest gaps, normalized by n / √ log n, they converge in ̷L p to a constant for all p> 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.
THÈSE DE DOCTORAT présentée par
, 2011
"... Etude des taux d’intérêt long terme Analyse stochastique des processus ponctuels déterminantaux Thèse présentée le 13/09/2010 devant le jury composé de: ..."
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Etude des taux d’intérêt long terme Analyse stochastique des processus ponctuels déterminantaux Thèse présentée le 13/09/2010 devant le jury composé de: