Determinantal probability measures

By Russell Lyons. 2002

284 citations recorded by the source corpus.

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.

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): “Many good mathematical surveys are now available [Borodin, 2009, Hough et al., 2006, Shirai and Takahashi, 2003a,b, Lyons, 2003, Soshnikov, 2000, Tao, 2009].”
Efficient Volume Sampling for Row/Column Subset Selection (2010, background): “row/column subset selection. Here are some ideas for future investigation suggested by this work: •It would be interesting to explore how these algorithmic ideas are related to determinantal sampling [16, 13] and, in particular, the generation of random spanning trees. •Find practical counterparts o”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “For further references and details on determinantal point processes we refer the reader to surveys [14, 17, 22, 37] and the lecture notes [38].”
The Palm measure and the Voronoi tessellation for the Ginibre process (2006, unknown): “The proof of Theorem 3 rests on an explicit description of the marginal laws of the process ψ , obtained in Section 1, which allows us to use a similar result proved by Lyons [19] (for commuting operators) and for Borcea, Branden and Liggett [2] (without this restriction) in the discrete determinant”
Hyperbolicity and stable polynomials in combinatorics and probability (2012, background): “Such measures were proved in [Lyo03] to be negatively associated.”
Determinantal processes and completeness of random exponentials: the critical case (2012, methodology): “Fourier conjugation, this gives rise to a non-negative contraction operator Q: ℓ2(Zd) → ℓ2(Zd). This, in turn, gives rise to a determinantal point process Pf on Zd in a canonical way, for details see [Ly03]. For a point conﬁguration ω drawn from the distribution of Pf, we denote by ω(k) the indicato”
Determinantal point processes (2018, background): “. Theorem 8.1 was proved in [Bur93]; another proof can be found in [Ben01]. The formulas for the ﬁrst and second correlation functions via Kgo back to [Kir1847] and [Bro40], respectively. We refer to [Lyo03] and references therein for further developments of the subject. Note that for planar graphs,”
Random Complexes and l^2-Betti Numbers (2008, methodology): “Most of our results were announced in Lyons (2003), Sec.”
On adding a list of numbers (and other one-dependent determinantal processes) (2009, background): “These include: the eigenvalues of various ensembles of random matrices (Dyson [26, 27, 28], Mehta [58]); the presence or absence of edges if a random spanning tree is chosen in a graph (Burton–Pemantle [11], Lyons [51]); random tilings and growth models (Johansson [47]); the structure of random part”
The Surprising Mathematics of Longest Increasing Subsequences (2015, unknown): “See [59], [80], [82], [81], [121] and [29, Section 5.”
Large time asymptotics of growth models on space-like paths I: PushASEP (2007, methodology): “Algebraically, we derive a determinantal formula for the distribution of the N -particle PushASEP with an arbitrary fixed initial condition, and we also represent this distribution as a gap probability for a (possibly, signed) determinantal point process (see [17; 12; 21; 16; 22] for information on ”
Determinantal Processes and Independence (2005, background): “Determinantal processes arise naturally in several other settings, including eigenvalues of random matrices, random spanning trees and nonintersecting paths; see, e.g., Burton and Pemantle [4], Soshnikov [27], Lyons [20], Lyons and Steif [21], Shirai and Takahashi [25], Johansson [14], Borodin, Okou”
Hyperdeterminantal point processes (2008, background): “minantalpointprocesseswereintroducedinMacchi(1975).Surveysoftheirproperties and numerous applications may be found in (Daley and Vere-Jones 1988; Soshnikov 2000; Lyons 2003; Hough et al. 2006; Shirai and Takahashi 2000, 2003a,b, 2004).”
Monte Carlo with determinantal point processes (2016, methodology): “For further information on determinantal point processes, we refer the reader to [Macchi, 1975, Hough et al., 2006, Johansson, 2006, Soshnikov, 2000, Lyons, 2003, Kulesza and Taskar, 2012, Lavancier et al., 2015].”
Unimodality, log-concavity, real-rootedness and beyond (2015, background): “PP matroid | is strong Rayleigh by the discussion in Section 9.1. A general class of strong Rayleigh measures containing the uniform spanning tree measures is the class of determinantal measures, see [66]. Let Cbe a hermitian n ncontraction matrix, i.e., a positive semidenite matrix with all its eig”
Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids (2006, background): “n property is known as negative association of the variables {Xe : e∈ E}, or of their partition function. The following important result is due to Feder and Mihail [16]; also see Theorem 6.5 of Lyons [26] and Theorem 1.3 of Pemantle [30]. Theorem3.8. If Z(ω;y) is Rayleigh and homogeneous then it is ”
Negative dependence and the geometry of polynomials (2007, background): “As we show in §3, this class contains several important examples such as distributions at time t ≥ 0 for symmetric exclusion processes generated by product or deterministic measures [55] (or, more generally, by strongly Rayleigh measures), and determinantal probability measures induced by positive c”