Janossy Densities. I. Determinantal Ensembles

By Alexei Borodin, Alexander Soshnikov. 2002

48 citations recorded by the source corpus.

We derive an elementary formula for Janossy densities for determinantal point processes with a finite rank projection-type kernel. In particular, for β=2 polynomial ensembles of random matrices we show that the Janossy densities on an interval I ⊂ ℝ can be expressed in terms of the Christoffel–Darboux kernel for the orthogonal polynomials on the complement of I.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Recently, DPPs have attracted a flurry of attention in the mathematics community [13, 14, 15, 16, 21, 72, 73, 74, 116, 117, 131], and much progress has been made in understanding their formal combinatorial and probabilistic properties.”
Near-Optimal MAP Inference for Determinantal Point Processes (2012, background): “Originally, DPPs were introduced to model fermions in quantum physics [8], but since then they have arisen in a variety of other settings including non-intersecting random paths, random spanning trees, and eigenvalues of random matrices [9, 23, 12].”
k-DPPs: Fixed-Size Determinantal Point Processes (2011, background): “Determinantal point processes , introduced to model fermions (Macchi, 1975), also arise in studies of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices (Daley & Vere-Jones, 2003; Borodin & Soshnikov , 2003; Hough et al., 2006).”
Gaussian fluctuations of eigenvalues in the GUE (2004, background): “. In [TW] the distribution of the largest eigenvalue was studied. Also eigenvalue number k with k ﬁxed has been studied. GAUSSIAN FLUCTUATIONS OF EIGENVALUES IN THE GUE 3 Remark 3. Set I = (−∞,s]. In [BS] it is shown that (1.3) P(xk ∈ (s +ds)) = 1 (k −1)! Z Ik−1 Jk(x1,...,xk−1,s)µ(dx1)...µ(dxk−1) ”
Structured Determinantal Point Processes (2010, background): “Determinantal point processes, introduced to model fermions [10], also arise in studies of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices [3, 2, 7].”