</script> We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region (D) is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on (L^2(D)). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.">
Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 3 (2006) open journal systems 


Determinantal processes and independence

J. Ben Hough, U.C. Berkeley
Manjunath Krishnapur, U.C. Berkeley
Yuval Peres, U.C. Berkeley
Balint Virag, University of Toronto


Abstract
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region \(D\) is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on \(L^2(D)\). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

Creative Common LOGO

Full Text: PDF
Hough, J. Ben, Krishnapur, Manjunath, Peres, Yuval, Virag, Balint, Determinantal processes and independence, Probability Surveys, 3, (2006), 206-229 (electronic).

References

[1]   Bapat, R.B. (1992). Mixed discriminants and spanning trees. Sankhya. Special Volume 54 49–55 MR1234678

[2]   Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29, 1–65. MR1825141

[3]   Borodin, A., Okounkov, A., Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13, 481–515. MR1758751

[4]   Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21, 1329–1371. MR1235419

[5]   Costin, O. and Lebowitz, J. (1995). Gaussian fluctuation in random matrices. Phys. Rev. Lett., 75 69–72.

[6]   Cox, D.R. (1955). Some statistical methods connected with series of events. J. R. Statist. Soc. B. 17 129–164. MR0092301

[7]   Daley, D.J. and Vere-Jones, D. (2003). An introduction to the theory of point processes. Vol. I. Elementary theory and Methods. Second edition. Springer-Verlag, New York. MR1950431

[8]   Diaconis, P. (2003). Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. 40 155–178. MR1962294

[9]   Gasiorowicz, S. (1996). Quantum Physics. John Wiley & Sons, Inc., New York, second edition.

[10]   Ginibre, J. (1965). Statistical ensembles of complex, quaternion and real matrices. J. Math. Phys. 6, 440–449. MR0173726

[11]   Goodman, N. and Dubman, M. (1969). Theory of time-varying spectral analysis and complex Wishart matrix processes. Multivariate Analysis, II (Proc. Second Internat. Sympos., Dayton, Ohio, 1968) pp. 351–366 Academic Press, New York MR0263205

[12]   Griffiths, R. C. (1984). Characterization of infinitely divisible multivariate gamma distributions. J. Multivariate Anal. 15, 13–20. MR0755813

[13]   Griffiths, R. C. and Milne, R. K. (1987). A class of infinitely divisible multivariate negative binomial distributions. J. Multivariate Anal. 22, 13–23. MR0890879

[14]   Johansson, K. (2004). Determinantal processes with number variance saturation. Comm. Math. Phys. 252 , 111–148. MR2103906

[15]   Karlin, S. and McGregor, J. (1959). Coincidence probabilities. Pacific J. Math. 9, 1141–1164. MR0114248

[16]   Khare, Avinash. (1997). Fractional statistics and quantum theory. World Scientific Publishing Co., Inc., River Edge, NJ. MR1795025

[17]   Kostlan, E. (1992). On the spectra of Gaussian matrices. Linear Algebra Appl., 162/164, 385–388. MR1148410

[18]   Lenard, A. (1973). Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys. 30, 35–44. MR0323270

[19]   Lenard, A. (1975). States of classical statistical mechanical systems of infinitely many particles. I-II. Arch. Rational Mech. Anal. 59, 219–256. MR0391830

[20]   Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212. MR2031202

[21]   Lyons, R. and Steif, J. (2003). Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120, 515–575. MR2030095

[22]   Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Probab., 7, 83–122. MR0380979

[23]   Peres, Y. and Virág, B. (2004). Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. Acta Math., to appear. arXiv:math.PR/0310297.

[24]   Rains, E. (1997). High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107, 219–241. MR1431220

[25]   Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants I: Fermion, Poisson and Boson processes. J. Funct. Analysis 205, 414–463. MR2018415

[26]   Shirai, T. and Takahashi, Y. (2003) Random point field associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties. Ann. Probab. 31, 1533–1564. MR1989442

[27]   Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys, 55, 923–975. MR1799012

[28]   Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab., 30, 171–187. MR1894104

[29]   Vere-Jones, D. (1997). Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math. 26, 125–149. MR1450811




Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787