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.
@article{PMIHES_2003__98__167_0,
author = {Lyons, Russell},
title = {Determinantal probability measures},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {167--212},
publisher = {Springer},
volume = {98},
year = {2003},
doi = {10.1007/s10240-003-0016-0},
mrnumber = {2031202},
zbl = {1055.60003},
language = {en},
url = {http://www.numdam.org/articles/10.1007/s10240-003-0016-0/}
}
Lyons, Russell. Determinantal probability measures. Publications Mathématiques de l'IHÉS, Tome 98 (2003), pp. 167-212. doi : 10.1007/s10240-003-0016-0. http://www.numdam.org/articles/10.1007/s10240-003-0016-0/
1. (1990), The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3, 450-465. | MR | Zbl
2. and (2001), The Probabilistic Method. Second edition. New York: John Wiley & Sons Inc. | MR | Zbl
3. , , , and (1999), Group-invariant percolation on graphs. Geom. Funct. Anal., 9, 29-66. | MR | Zbl
4. , , , and (2001), Uniform spanning forests. Ann. Probab., 29, 1-65. | MR | Zbl
5. , and (1985), Inequalities with applications to percolation and reliability. J. Appl. Probab., 22, 556-569. | MR | Zbl
6. and (1967), On the closure of characters and the zeros of entire functions. Acta Math., 118, 79-93. | MR | Zbl
7. (2000), Characters of symmetric groups, and correlation functions of point processes. Funkts. Anal. Prilozh., 34, 12-28, 96. English translation: Funct. Anal. Appl., 34(1), 10-23. | MR | Zbl
8. , , and (2000), Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc., 13, 481-515 (electronic). | MR | Zbl
9. and (2000), Distributions on partitions, point processes, and the hypergeometric kernel. Comment. Math. Phys., 211, 335-358. | MR | Zbl
10. and (2001), z-measures on partitions, Robinson-Schensted-Knuth correspondence, and β=2 random matrix ensembles. In P. Bleher and A. Its, eds., Random Matrix Models and Their Applications, vol. 40 of Math. Sci. Res. Inst. Publ., pp. 71-94. Cambridge: Cambridge Univ. Press. | Zbl
11. and (2002), Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes. Preprint. | MR | Zbl
12. and (1987), Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis. Isr. J. Math., 57, 137-224. | Zbl
13. A. Broder (1989), Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, North Carolina), pp. 442-447. New York: IEEE.
14. , , , and (1940), The dissection of rectangles into squares. Duke Math. J., 7, 312-340. | MR | Zbl
15. and (1993), Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab., 21, 1329-1371. | MR | Zbl
16. and (1986), L2-cohomology and group cohomology. Topology, 25, 189-215. | MR | Zbl
17. , , , and (2003), Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. To appear. | MR | Zbl
18. (2003), The Riemann hypothesis. Notices Am. Math. Soc., 50, 341-353. | MR
19. (1990), A Course in Functional Analysis. Second edition. New York: Springer. | MR | Zbl
20. (1972/73), Entropie d'un groupe abélien de transformations. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 25, 11-30. | Zbl
21. and (1988), An Introduction to the Theory of Point Processes. New York: Springer. | MR | Zbl
22. (2003), Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Am. Math. Soc., New Ser., 40, 155-178 (electronic). | MR
23. and (1998), Balls and bins: a study in negative dependence. Random Struct. Algorithms, 13, 99-124. | MR | Zbl
24. (1962), Statistical theory of the energy levels of complex systems. III. J. Math. Phys., 3, 166-175. | MR | Zbl
25. T. Feder and M. Mihail (1992), Balanced matroids. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 26-38, New York. Association for Computing Machinery (ACM). Held in Victoria, BC, Canada.
26. (1948), The average impedance of an electrical network. In Reissner Anniversary Volume, Contributions to Applied Mechanics, pp. 333-340. J. W. Edwards, Ann Arbor, Michigan. Edited by the Staff of the Department of Aeronautical Engineering and Applied Mechanics of the Polytechnic Institute of Brooklyn. | MR | Zbl
27. and (1991), Representation Theory: A First Course. Readings in Mathematics. New York: Springer. | MR | Zbl
28. (2002), Invariants l2 de relations d'équivalence et de groupes. Publ. Math., Inst. Hautes Étud. Sci., 95, 93-150. | Numdam | Zbl
29. (1988), Gibbs Measures and Phase Transitions. Berlin-New York: Walter de Gruyter & Co. | MR | Zbl
30. (1995), Random-cluster measures and uniform spanning trees. Stochastic Processes Appl., 59, 267-275. | MR | Zbl
31. (1982), A Hilbert Space Problem Book. Second edition. Encycl. Math. Appl. 17, New York: Springer. | MR | Zbl
32. and (2003), Change intolerance in spanning forests. J. Theor. Probab., 16, 47-58. | MR | Zbl
33. (2001), Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. (2), 153, 259-296. | MR | Zbl
34. (2002), Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields, 123, 225-280. | MR | Zbl
35. (1983), Enumeration of Q-acyclic simplicial complexes. Isr. J. Math., 45, 337-351. | MR | Zbl
36. and (1972), Commuting measure-preserving transformations. Isr. J. Math., 12, 161-173. | MR | Zbl
37. G. Kirchhoff (1847), Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem., 72, 497-508.
38. (1998), A bird's-eye view of uniform spanning trees and forests. In D. Aldous and J. Propp, eds., Microsurveys in Discrete Probability, vol. 41 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 135-162. Providence, RI: Am. Math. Soc., Papers from the workshop held as part of the Dimacs Special Year on Discrete Probability in Princeton, NJ, June 2-6, 1997. | Zbl
39. (2000), Phase transitions on nonamenable graphs. J. Math. Phys., 41, 1099-1126. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. | MR | Zbl
40. R. Lyons (2003), Random complexes and ℓ2-Betti numbers. In preparation.
41. , , and (2003), Minimal spanning forests. In preparation.
42. and (2003), Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. To appear. | MR | Zbl
43. (1975), The coincidence approach to stochastic point processes. Adv. Appl. Probab., 7, 83-122. | MR | Zbl
44. (1976), Matrix generalizations of some theorems on trees, cycles and cocycles in graphs. SIAM J. Appl. Math., 30, 143-148. | MR | Zbl
45. (1991), Random Matrices. Second edition. Boston, MA: Academic Press Inc. | MR | Zbl
46. (2003), The components of the wired spanning forest are recurrent. Probab. Theory Related Fields, 125, 259-265. | MR | Zbl
47. C. M. Newman (1984), Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y. L. Tong, ed., Inequalities in Statistics and Probability, pp. 127-140. Hayward, CA: Inst. Math. Statist. Proceedings of the symposium held at the University of Nebraska, Lincoln, Neb., October 27-30, 1982. | MR
48. (2001), Infinite wedge and random partitions. Sel. Math., New Ser., 7, 57-81. | MR | Zbl
49. and (2003), Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc., 16, 581-603 (electronic). | MR | Zbl
50. and (1987), Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math., 48, 1-141. | MR | Zbl
51. (1992), Matroid Theory. New York: Oxford University Press. | MR | Zbl
52. (1991), Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19, 1559-1574. | MR | Zbl
53. (2000), Towards a theory of negative dependence. J. Math. Phys., 41, 1371-1390. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. | MR | Zbl
54. and (1998), How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms, 27, 170-217. 7th Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 1996). | MR | Zbl
55. (1972), Two consequences of the Beurling-Malliavin theory. Proc. Am. Math. Soc., 36, 116-122. | MR | Zbl
56. (1977), Completeness of sets of complex exponentials. Adv. Math., 24, 1-62. | MR | Zbl
57. and (1997), The Beurling-Malliavin density of a random sequence. Proc. Am. Math. Soc., 125, 1745-1749. | MR | Zbl
58. (2000), A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab., 13, 343-356. | MR | Zbl
59. and (1999), The law of the iterated logarithm for negatively associated random variables. Stochastic Processes Appl., 83, 139-148. | MR | Zbl
60. and (2000), Fermion process and Fredholm determinant. In H. G. W. Begehr, R. P. Gilbert, and J. Kajiwara, eds., Proceedings of the Second ISAAC Congress, vol. 1, pp. 15-23. Kluwer Academic Publ. International Society for Analysis, Applications and Computation, vol. 7. | MR | Zbl
61. and (2002), Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. Preprint. | MR | Zbl
62. and (2003), Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic and Gibbs properties. Ann. Probab., 31, 1533-1564. | MR | Zbl
63. and (2002), Glauber dynamics for fermion point processes. Nagoya Math. J., 168, 139-166. | MR | Zbl
64. (2000a), Determinantal random point fields. Usp. Mat. Nauk, 55, 107-160. | MR | Zbl
65. (2000b), Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Stat. Phys., 100, 491-522. | MR | Zbl
66. (1965), The existence of probability measures with given marginals. Ann. Math. Stat., 36, 423-439. | MR | Zbl
67. (1990), Resistances and currents in infinite electrical networks. J. Combin. Theory, Ser. B, 49, 87-102. | MR | Zbl
68. (1972), Convergence en moyenne de l'information pour l'action de Z2. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 24, 135-137. | Zbl
69. and (1981), Asymptotic theory of the characters of a symmetric group. Funkts. Anal. i Prilozh., 15, 15-27, 96. English translation: Funct. Anal. Appl., 15(4), 246-255 (1982). | MR | Zbl
70. (1976), Matroid Theory. London: Academic Press [Harcourt Brace Jovanovich Publishers]. L. M. S. Monographs, No. 8. | MR | Zbl
71. , ed. (1987), Combinatorial Geometries. Cambridge: Cambridge University Press. | MR | Zbl
72. (1935), On the abstract properties of linear dependence. Am. J. Math., 57, 509-533. | MR | Zbl
73. (1957), Geometric Integration Theory. Princeton, N.J.: Princeton University Press. | MR | Zbl
74. (1996), Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing, pp. 296-303. New York: ACM. Held in Philadelphia, PA, May 22-24, 1996. | MR | Zbl
75. (2001), Strassen's law of the iterated logarithm for negatively associated random vectors. Stochastic Processes Appl., 95, 311-328. | Zbl
76. and (2001), A weak convergence for negatively associated fields. Stat. Probab. Lett., 53, 259-267. | MR | Zbl
Cité par Sources :
