## Random point fields associated with certain Fredholm determinants. II: Fermion shifts and their ergodic and Gibbs properties.(English)Zbl 1051.60053

[For Part I, J. Funct. Anal. 205, No. 2, 414–463 (2003; Zbl 1051.60052), see above.]
Random point fields form an interesting class of random measures and have been studied from many angles. Processes that do not admit multiple points are quite amenable for the study of Fermion point processes. The authors of the present paper focus their attention on the case when $$R$$ is a countable space and $$\lambda$$ is the counting measure with $$L^{2}(R,\lambda)=\ell^{2}(R).$$ They construct a family of probability measures on the configuration space over countable discrete space associated with nonnegative symmetric operators via determinants. Using this theory they establish that
(i) the measure structure is discrete and determinantal;
(ii) the metric entropy of a Fermion shift is positive except in trivial cases when the operator in question is zero or identity; and
(iii) the Fermion process is tail trivial under certain mild constraints on the measure.
Besides the paper also deals with the ergodic properties of the process in question.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures 60G60 Random fields 82B10 Quantum equilibrium statistical mechanics (general) 28D20 Entropy and other invariants

### Keywords:

point process; Fermion point process; Gibbs measure

Zbl 1051.60052
Full Text:

### References:

 [1] BORODIN, A., OKOUNKOV, A. and OLSHANSKI, G. (2000). Asy mptotics of Plancherel measures for sy mmetric groups. J. Amer. Math. Soc. 13 481-515. JSTOR: · Zbl 0938.05061 [2] BORODIN, A. and OLSHANSKI, G. (1998). Point processes and the infinite sy mmetric group, III: Fermion point processes. Available at xxx.lanl.gov/abs/math.RT/9804088. [3] CORNFELD, I. P., FOMIN, S. V. and SINAI, YA. G. (1982). Introduction to Ergodic Theory. Springer, New York. · Zbl 0493.28007 [4] COSTIN, O. and LEBOWITZ, J. (1995). Gaussian fluctuation in random matrices. Phy s. Rev. Lett. 75 69-72. [5] DALEY, D. J. and VERES-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. · Zbl 0657.60069 [6] Dy M, H. and MCKEAN, H. P. (1972). Fourier Series and Integrals. Academic Press, New York. · Zbl 0242.42001 [7] GEORGII, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruy ter, Berlin. · Zbl 0657.60122 [8] GOHBERG, I. C. and KREIN, M. G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators. Amer. Math. Soc., Providence, RI. · Zbl 0181.13504 [9] GRENANDER, U. and SZEGÖ, G. (1984). Toeplitz Forms and Their Applications, 2nd ed. Chelsea, New York. · Zbl 0611.47018 [10] ITO, S. and TAKAHASHI, Y. (1974). Markov subshifts and realization of -expansions. J. Math. Soc. Japan 26 33-55. · Zbl 0269.28006 [11] KAC, M. (1954). Toeplitz matrices, translation kernel and a related problem in probability theory. Duke Math. J. 21 501-509. · Zbl 0056.10201 [12] LIGGETT, T. M. (1985). Interacting Particle Sy stems. Springer, New York. [13] MACCHI, O. (1974). The fermion process-a model of stochastic point process with repulsive points. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians A 391-398. Reidel, Dordrecht. · Zbl 0413.60048 [14] MACCHI, O. (1975). The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 83-122. JSTOR: · Zbl 0366.60081 [15] MEHTA, M. L. (1991). Random Matrices, 2nd ed. Academic Press, New York. · Zbl 0780.60014 [16] SHIGA, T. (1977). Some problems related to Gibbs states, canonical Gibbs states and Markovian time evolutions. Z. Wahrsch. Verw. Gebiete 39 339-352. · Zbl 0384.60075 [17] SHIRAI, T. and TAKAHASHI, Y. (2000). Fermion process and Fredholm determinant. In Proceedings of the Second ISAAC Congress (H. G. W. Begehr, R. P. Gilbert and J. Kajiwara, eds.) 1 15-23. Kluwer, Dordrecht. · Zbl 1036.60045 [18] SHIRAI, T. and TAKAHASHI, Y. (2002). Random point fields associated with certain Fredholm determinants, I: Fermion, Poisson, and boson point processes. · Zbl 1051.60052 [19] SHIRAI, T. and YOO, H. J. (2002). Glauber dy namics for fermion point processes. Nagoy a Math. J. 168 139-166. · Zbl 1029.82025 [20] SIMON, B. (1977). Trace Ideals and Their Applications. Cambridge Univ. Press. [21] SOSHNIKOV, A. (2000). Gaussian fluctuation for the number of particles in Airy, Bessel, sine and other determinantal random point fields. J. Statist. Phy s. 100 491-522. · Zbl 1041.82001 [22] SOSHNIKOV, A. (2000). Determinantal random point fields. Russian Math. Survey s 55 923-975. · Zbl 0991.60038 [23] SOSHNIKOV, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30 171-187. · Zbl 1033.60063 [24] SPOHN, H. (1987). Interacting Brownian Particles: A Study of Dy son’s Model. In Hy drody namic Behavior and Interacting Particle Sy stems (G. Papanicalaou, ed.) 151-179. Springer, New York. · Zbl 0674.60096 [25] SZEGÖ, G. (1920). Beiträge zur Theorie der Toeplitzchen Formen (Erste Mittelung). Math. Z. 6 167-202. · JFM 47.0391.04 [26] SZEGÖ, G. (1959). Orthogonal Poly nomials. Amer. Math. Soc., New York. [27] TOTOKI, H. (1971). Introduction to Ergodic Theory. Ky oritsu shuppan, Toky o. (In Japanese.) [28] KANAZAWA, ISHIKAWA 920-1192 JAPAN E-MAIL: shirai@kenroku.kanazawa-u.ac.jp RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES Ky OTO UNIVERSITY SAKy O-KU, Ky OTO 606-8502 JAPAN E-MAIL: takahasi@kurims.ky oto-u.ac.jp
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